| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eqid 2736 | . . . . . . . . 9
⊢ (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) = (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) | 
| 2 | 1 | elrnmpt 5968 | . . . . . . . 8
⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ↔ ∃𝑦 ∈ 𝐵 𝑧 = (◡𝐹 “ {𝑦}))) | 
| 3 | 2 | elv 3484 | . . . . . . 7
⊢ (𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ↔ ∃𝑦 ∈ 𝐵 𝑧 = (◡𝐹 “ {𝑦})) | 
| 4 |  | simpr 484 | . . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (◡𝐹 “ {𝑦})) → 𝑧 = (◡𝐹 “ {𝑦})) | 
| 5 |  | simpl3 1193 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → 𝐵 ⊆ ran 𝐹) | 
| 6 |  | simpr 484 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵) | 
| 7 | 5, 6 | sseldd 3983 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ran 𝐹) | 
| 8 |  | inisegn0 6115 | . . . . . . . . . . 11
⊢ (𝑦 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑦}) ≠ ∅) | 
| 9 | 7, 8 | sylib 218 | . . . . . . . . . 10
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → (◡𝐹 “ {𝑦}) ≠ ∅) | 
| 10 | 9 | adantr 480 | . . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (◡𝐹 “ {𝑦})) → (◡𝐹 “ {𝑦}) ≠ ∅) | 
| 11 | 4, 10 | eqnetrd 3007 | . . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (◡𝐹 “ {𝑦})) → 𝑧 ≠ ∅) | 
| 12 | 11 | r19.29an 3157 | . . . . . . 7
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ ∃𝑦 ∈ 𝐵 𝑧 = (◡𝐹 “ {𝑦})) → 𝑧 ≠ ∅) | 
| 13 | 3, 12 | sylan2b 594 | . . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) → 𝑧 ≠ ∅) | 
| 14 | 13 | ralrimiva 3145 | . . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))𝑧 ≠ ∅) | 
| 15 |  | simp2 1137 | . . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐹 Fn 𝐴) | 
| 16 |  | simp1 1136 | . . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐴 ∈ 𝑉) | 
| 17 | 15, 16 | jca 511 | . . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉)) | 
| 18 |  | fnex 7238 | . . . . . . . 8
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | 
| 19 |  | rnexg 7925 | . . . . . . . 8
⊢ (𝐹 ∈ V → ran 𝐹 ∈ V) | 
| 20 | 17, 18, 19 | 3syl 18 | . . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ran 𝐹 ∈ V) | 
| 21 |  | simp3 1138 | . . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐵 ⊆ ran 𝐹) | 
| 22 | 20, 21 | ssexd 5323 | . . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐵 ∈ V) | 
| 23 |  | mptexg 7242 | . . . . . 6
⊢ (𝐵 ∈ V → (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V) | 
| 24 |  | rnexg 7925 | . . . . . 6
⊢ ((𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V → ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V) | 
| 25 |  | fvi 6984 | . . . . . 6
⊢ (ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V → ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) = ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 26 | 22, 23, 24, 25 | 4syl 19 | . . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) = ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 27 | 14, 26 | raleqtrrdv 3329 | . . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∀𝑧 ∈ ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))𝑧 ≠ ∅) | 
| 28 |  | fvex 6918 | . . . . 5
⊢ ( I
‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∈ V | 
| 29 | 28 | ac5b 10519 | . . . 4
⊢
(∀𝑧 ∈ (
I ‘ran (𝑦 ∈
𝐵 ↦ (◡𝐹 “ {𝑦})))𝑧 ≠ ∅ → ∃𝑓(𝑓:( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))⟶∪ ( I
‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))(𝑓‘𝑧) ∈ 𝑧)) | 
| 30 | 27, 29 | syl 17 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∃𝑓(𝑓:( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))⟶∪ ( I
‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))(𝑓‘𝑧) ∈ 𝑧)) | 
| 31 | 26 | unieqd 4919 | . . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪ ( I
‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) = ∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 32 | 26, 31 | feq23d 6730 | . . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (𝑓:( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))⟶∪ ( I
‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ↔ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))) | 
| 33 | 26 | raleqdv 3325 | . . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (∀𝑧 ∈ ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))(𝑓‘𝑧) ∈ 𝑧 ↔ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧)) | 
| 34 | 32, 33 | anbi12d 632 | . . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ((𝑓:( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))⟶∪ ( I
‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))(𝑓‘𝑧) ∈ 𝑧) ↔ (𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧))) | 
| 35 | 34 | exbidv 1920 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (∃𝑓(𝑓:( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))⟶∪ ( I
‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))(𝑓‘𝑧) ∈ 𝑧) ↔ ∃𝑓(𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧))) | 
| 36 | 30, 35 | mpbid 232 | . 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∃𝑓(𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧)) | 
| 37 |  | vex 3483 | . . . . . . . . 9
⊢ 𝑓 ∈ V | 
| 38 | 37 | rnex 7933 | . . . . . . . 8
⊢ ran 𝑓 ∈ V | 
| 39 | 38 | a1i 11 | . . . . . . 7
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ran 𝑓 ∈ V) | 
| 40 |  | simplr 768 | . . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 41 |  | frn 6742 | . . . . . . . . 9
⊢ (𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → ran 𝑓 ⊆ ∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 42 | 40, 41 | syl 17 | . . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ran 𝑓 ⊆ ∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 43 |  | nfv 1913 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑦(𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) | 
| 44 |  | nfcv 2904 | . . . . . . . . . . . . . 14
⊢
Ⅎ𝑦𝑓 | 
| 45 |  | nfmpt1 5249 | . . . . . . . . . . . . . . 15
⊢
Ⅎ𝑦(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) | 
| 46 | 45 | nfrn 5962 | . . . . . . . . . . . . . 14
⊢
Ⅎ𝑦ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) | 
| 47 | 46 | nfuni 4913 | . . . . . . . . . . . . . 14
⊢
Ⅎ𝑦∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) | 
| 48 | 44, 46, 47 | nff 6731 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑦 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) | 
| 49 | 43, 48 | nfan 1898 | . . . . . . . . . . . 12
⊢
Ⅎ𝑦((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 50 |  | nfv 1913 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑦(𝑓‘𝑧) ∈ 𝑧 | 
| 51 | 46, 50 | nfralw 3310 | . . . . . . . . . . . 12
⊢
Ⅎ𝑦∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧 | 
| 52 | 49, 51 | nfan 1898 | . . . . . . . . . . 11
⊢
Ⅎ𝑦(((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) | 
| 53 | 17, 18 | syl 17 | . . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐹 ∈ V) | 
| 54 | 53 | ad3antrrr 730 | . . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) → 𝐹 ∈ V) | 
| 55 |  | cnvexg 7947 | . . . . . . . . . . . . . 14
⊢ (𝐹 ∈ V → ◡𝐹 ∈ V) | 
| 56 |  | imaexg 7936 | . . . . . . . . . . . . . 14
⊢ (◡𝐹 ∈ V → (◡𝐹 “ {𝑦}) ∈ V) | 
| 57 | 54, 55, 56 | 3syl 18 | . . . . . . . . . . . . 13
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) → (◡𝐹 “ {𝑦}) ∈ V) | 
| 58 |  | cnvimass 6099 | . . . . . . . . . . . . . . 15
⊢ (◡𝐹 “ {𝑦}) ⊆ dom 𝐹 | 
| 59 | 58 | a1i 11 | . . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) → (◡𝐹 “ {𝑦}) ⊆ dom 𝐹) | 
| 60 | 15 | fndmd 6672 | . . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → dom 𝐹 = 𝐴) | 
| 61 | 60 | ad3antrrr 730 | . . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) → dom 𝐹 = 𝐴) | 
| 62 | 59, 61 | sseqtrd 4019 | . . . . . . . . . . . . 13
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) → (◡𝐹 “ {𝑦}) ⊆ 𝐴) | 
| 63 | 57, 62 | elpwd 4605 | . . . . . . . . . . . 12
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) → (◡𝐹 “ {𝑦}) ∈ 𝒫 𝐴) | 
| 64 | 63 | ex 412 | . . . . . . . . . . 11
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑦 ∈ 𝐵 → (◡𝐹 “ {𝑦}) ∈ 𝒫 𝐴)) | 
| 65 | 52, 64 | ralrimi 3256 | . . . . . . . . . 10
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ∀𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦}) ∈ 𝒫 𝐴) | 
| 66 | 1 | rnmptss 7142 | . . . . . . . . . 10
⊢
(∀𝑦 ∈
𝐵 (◡𝐹 “ {𝑦}) ∈ 𝒫 𝐴 → ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ⊆ 𝒫 𝐴) | 
| 67 | 65, 66 | syl 17 | . . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ⊆ 𝒫 𝐴) | 
| 68 |  | sspwuni 5099 | . . . . . . . . 9
⊢ (ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ⊆ 𝒫 𝐴 ↔ ∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ⊆ 𝐴) | 
| 69 | 67, 68 | sylib 218 | . . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ⊆ 𝐴) | 
| 70 | 42, 69 | sstrd 3993 | . . . . . . 7
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ran 𝑓 ⊆ 𝐴) | 
| 71 | 39, 70 | elpwd 4605 | . . . . . 6
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ran 𝑓 ∈ 𝒫 𝐴) | 
| 72 |  | fnfun 6667 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | 
| 73 | 15, 72 | syl 17 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → Fun 𝐹) | 
| 74 | 73 | ad5antr 734 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ (𝑓‘𝑢) = (𝑓‘𝑣)) → Fun 𝐹) | 
| 75 |  | sndisj 5134 | . . . . . . . . . . . . . . . . . . 19
⊢
Disj 𝑦 ∈
𝐵 {𝑦} | 
| 76 |  | disjpreima 32598 | . . . . . . . . . . . . . . . . . . 19
⊢ ((Fun
𝐹 ∧ Disj 𝑦 ∈ 𝐵 {𝑦}) → Disj 𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦})) | 
| 77 | 74, 75, 76 | sylancl 586 | . . . . . . . . . . . . . . . . . 18
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ (𝑓‘𝑢) = (𝑓‘𝑣)) → Disj 𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦})) | 
| 78 |  | disjrnmpt 32599 | . . . . . . . . . . . . . . . . . 18
⊢
(Disj 𝑦
∈ 𝐵 (◡𝐹 “ {𝑦}) → Disj 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))𝑧) | 
| 79 | 77, 78 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ (𝑓‘𝑢) = (𝑓‘𝑣)) → Disj 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))𝑧) | 
| 80 |  | simpllr 775 | . . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ (𝑓‘𝑢) = (𝑓‘𝑣)) → 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 81 |  | simplr 768 | . . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ (𝑓‘𝑢) = (𝑓‘𝑣)) → 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 82 |  | simp-4r 783 | . . . . . . . . . . . . . . . . . 18
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ (𝑓‘𝑢) = (𝑓‘𝑣)) → ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) | 
| 83 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = 𝑢 → (𝑓‘𝑧) = (𝑓‘𝑢)) | 
| 84 |  | id 22 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = 𝑢 → 𝑧 = 𝑢) | 
| 85 | 83, 84 | eleq12d 2834 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 𝑢 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑢)) | 
| 86 | 85 | rspcv 3617 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧 → (𝑓‘𝑢) ∈ 𝑢)) | 
| 87 | 86 | imp 406 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑓‘𝑢) ∈ 𝑢) | 
| 88 | 80, 82, 87 | syl2anc 584 | . . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ (𝑓‘𝑢) = (𝑓‘𝑣)) → (𝑓‘𝑢) ∈ 𝑢) | 
| 89 |  | simpr 484 | . . . . . . . . . . . . . . . . . 18
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ (𝑓‘𝑢) = (𝑓‘𝑣)) → (𝑓‘𝑢) = (𝑓‘𝑣)) | 
| 90 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 = 𝑣 → (𝑓‘𝑧) = (𝑓‘𝑣)) | 
| 91 |  | id 22 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 = 𝑣 → 𝑧 = 𝑣) | 
| 92 | 90, 91 | eleq12d 2834 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = 𝑣 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑣) ∈ 𝑣)) | 
| 93 | 92 | rspcv 3617 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧 → (𝑓‘𝑣) ∈ 𝑣)) | 
| 94 | 93 | imp 406 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑓‘𝑣) ∈ 𝑣) | 
| 95 | 81, 82, 94 | syl2anc 584 | . . . . . . . . . . . . . . . . . 18
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ (𝑓‘𝑢) = (𝑓‘𝑣)) → (𝑓‘𝑣) ∈ 𝑣) | 
| 96 | 89, 95 | eqeltrd 2840 | . . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ (𝑓‘𝑢) = (𝑓‘𝑣)) → (𝑓‘𝑢) ∈ 𝑣) | 
| 97 | 84, 91 | disji 5127 | . . . . . . . . . . . . . . . . 17
⊢
((Disj 𝑧
∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))𝑧 ∧ (𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ((𝑓‘𝑢) ∈ 𝑢 ∧ (𝑓‘𝑢) ∈ 𝑣)) → 𝑢 = 𝑣) | 
| 98 | 79, 80, 81, 88, 96, 97 | syl122anc 1380 | . . . . . . . . . . . . . . . 16
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ (𝑓‘𝑢) = (𝑓‘𝑣)) → 𝑢 = 𝑣) | 
| 99 | 98 | ex 412 | . . . . . . . . . . . . . . 15
⊢
((((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) → ((𝑓‘𝑢) = (𝑓‘𝑣) → 𝑢 = 𝑣)) | 
| 100 | 99 | anasss 466 | . . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ (𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ 𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))) → ((𝑓‘𝑢) = (𝑓‘𝑣) → 𝑢 = 𝑣)) | 
| 101 | 100 | ralrimivva 3201 | . . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ∀𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))∀𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))((𝑓‘𝑢) = (𝑓‘𝑣) → 𝑢 = 𝑣)) | 
| 102 | 40, 101 | jca 511 | . . . . . . . . . . . 12
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))∀𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))((𝑓‘𝑢) = (𝑓‘𝑣) → 𝑢 = 𝑣))) | 
| 103 |  | dff13 7276 | . . . . . . . . . . . 12
⊢ (𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))–1-1→∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ↔ (𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))∀𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))((𝑓‘𝑢) = (𝑓‘𝑣) → 𝑢 = 𝑣))) | 
| 104 | 102, 103 | sylibr 234 | . . . . . . . . . . 11
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))–1-1→∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 105 |  | f1f1orn 6858 | . . . . . . . . . . 11
⊢ (𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))–1-1→∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))–1-1-onto→ran
𝑓) | 
| 106 | 104, 105 | syl 17 | . . . . . . . . . 10
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))–1-1-onto→ran
𝑓) | 
| 107 |  | f1oen3g 9008 | . . . . . . . . . 10
⊢ ((𝑓 ∈ V ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))–1-1-onto→ran
𝑓) → ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ≈ ran 𝑓) | 
| 108 | 37, 106, 107 | sylancr 587 | . . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ≈ ran 𝑓) | 
| 109 | 108 | ensymd 9046 | . . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ran 𝑓 ≈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 110 | 22, 23 | syl 17 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V) | 
| 111 | 110 | ad2antrr 726 | . . . . . . . . . 10
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V) | 
| 112 | 57 | ex 412 | . . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑦 ∈ 𝐵 → (◡𝐹 “ {𝑦}) ∈ V)) | 
| 113 | 52, 112 | ralrimi 3256 | . . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ∀𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦}) ∈ V) | 
| 114 | 73 | ad5antr 734 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑦 ≠ 𝑡) → Fun 𝐹) | 
| 115 |  | simpr 484 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑦 ≠ 𝑡) → 𝑦 ≠ 𝑡) | 
| 116 | 21 | ad5antr 734 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑦 ≠ 𝑡) → 𝐵 ⊆ ran 𝐹) | 
| 117 |  | simpllr 775 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑦 ≠ 𝑡) → 𝑦 ∈ 𝐵) | 
| 118 | 116, 117 | sseldd 3983 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑦 ≠ 𝑡) → 𝑦 ∈ ran 𝐹) | 
| 119 |  | simplr 768 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑦 ≠ 𝑡) → 𝑡 ∈ 𝐵) | 
| 120 | 116, 119 | sseldd 3983 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑦 ≠ 𝑡) → 𝑡 ∈ ran 𝐹) | 
| 121 | 114, 115,
118, 120 | preimane 32681 | . . . . . . . . . . . . . . . . . 18
⊢
(((((((𝐴 ∈
𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑦 ≠ 𝑡) → (◡𝐹 “ {𝑦}) ≠ (◡𝐹 “ {𝑡})) | 
| 122 | 121 | ex 412 | . . . . . . . . . . . . . . . . 17
⊢
((((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → (𝑦 ≠ 𝑡 → (◡𝐹 “ {𝑦}) ≠ (◡𝐹 “ {𝑡}))) | 
| 123 | 122 | necon4d 2963 | . . . . . . . . . . . . . . . 16
⊢
((((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → ((◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑡}) → 𝑦 = 𝑡)) | 
| 124 | 123 | ralrimiva 3145 | . . . . . . . . . . . . . . 15
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑦 ∈ 𝐵) → ∀𝑡 ∈ 𝐵 ((◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑡}) → 𝑦 = 𝑡)) | 
| 125 | 124 | ex 412 | . . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑦 ∈ 𝐵 → ∀𝑡 ∈ 𝐵 ((◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑡}) → 𝑦 = 𝑡))) | 
| 126 | 52, 125 | ralrimi 3256 | . . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ∀𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐵 ((◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑡}) → 𝑦 = 𝑡)) | 
| 127 | 113, 126 | jca 511 | . . . . . . . . . . . 12
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦}) ∈ V ∧ ∀𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐵 ((◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑡}) → 𝑦 = 𝑡))) | 
| 128 |  | sneq 4635 | . . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑡 → {𝑦} = {𝑡}) | 
| 129 | 128 | imaeq2d 6077 | . . . . . . . . . . . . 13
⊢ (𝑦 = 𝑡 → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑡})) | 
| 130 | 1, 129 | f1mpt 7282 | . . . . . . . . . . . 12
⊢ ((𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})):𝐵–1-1→V ↔ (∀𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦}) ∈ V ∧ ∀𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐵 ((◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑡}) → 𝑦 = 𝑡))) | 
| 131 | 127, 130 | sylibr 234 | . . . . . . . . . . 11
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})):𝐵–1-1→V) | 
| 132 |  | f1f1orn 6858 | . . . . . . . . . . 11
⊢ ((𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})):𝐵–1-1→V → (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})):𝐵–1-1-onto→ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 133 | 131, 132 | syl 17 | . . . . . . . . . 10
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})):𝐵–1-1-onto→ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 134 |  | f1oen3g 9008 | . . . . . . . . . 10
⊢ (((𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V ∧ (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})):𝐵–1-1-onto→ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) → 𝐵 ≈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 135 | 111, 133,
134 | syl2anc 584 | . . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → 𝐵 ≈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 136 | 135 | ensymd 9046 | . . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ≈ 𝐵) | 
| 137 |  | entr 9047 | . . . . . . . 8
⊢ ((ran
𝑓 ≈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ≈ 𝐵) → ran 𝑓 ≈ 𝐵) | 
| 138 | 109, 136,
137 | syl2anc 584 | . . . . . . 7
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ran 𝑓 ≈ 𝐵) | 
| 139 |  | imass2 6119 | . . . . . . . . . . 11
⊢ (ran
𝑓 ⊆ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (𝐹 “ ran 𝑓) ⊆ (𝐹 “ ∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))) | 
| 140 | 41, 139 | syl 17 | . . . . . . . . . 10
⊢ (𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (𝐹 “ ran 𝑓) ⊆ (𝐹 “ ∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))) | 
| 141 | 40, 140 | syl 17 | . . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝐹 “ ran 𝑓) ⊆ (𝐹 “ ∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))) | 
| 142 |  | imauni 7267 | . . . . . . . . . 10
⊢ (𝐹 “ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) = ∪
𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝐹 “ 𝑧) | 
| 143 |  | imaeq2 6073 | . . . . . . . . . . . . 13
⊢ (𝑧 = (◡𝐹 “ {𝑦}) → (𝐹 “ 𝑧) = (𝐹 “ (◡𝐹 “ {𝑦}))) | 
| 144 | 53 | adantr 480 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → 𝐹 ∈ V) | 
| 145 | 144, 55, 56 | 3syl 18 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → (◡𝐹 “ {𝑦}) ∈ V) | 
| 146 | 143, 145 | iunrnmptss 32579 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪
𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝐹 “ 𝑧) ⊆ ∪
𝑦 ∈ 𝐵 (𝐹 “ (◡𝐹 “ {𝑦}))) | 
| 147 |  | funimacnv 6646 | . . . . . . . . . . . . . . . . 17
⊢ (Fun
𝐹 → (𝐹 “ (◡𝐹 “ {𝑦})) = ({𝑦} ∩ ran 𝐹)) | 
| 148 | 73, 147 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (𝐹 “ (◡𝐹 “ {𝑦})) = ({𝑦} ∩ ran 𝐹)) | 
| 149 | 148 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → (𝐹 “ (◡𝐹 “ {𝑦})) = ({𝑦} ∩ ran 𝐹)) | 
| 150 | 6 | snssd 4808 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → {𝑦} ⊆ 𝐵) | 
| 151 | 150, 5 | sstrd 3993 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → {𝑦} ⊆ ran 𝐹) | 
| 152 |  | dfss2 3968 | . . . . . . . . . . . . . . . 16
⊢ ({𝑦} ⊆ ran 𝐹 ↔ ({𝑦} ∩ ran 𝐹) = {𝑦}) | 
| 153 | 151, 152 | sylib 218 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → ({𝑦} ∩ ran 𝐹) = {𝑦}) | 
| 154 | 149, 153 | eqtrd 2776 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → (𝐹 “ (◡𝐹 “ {𝑦})) = {𝑦}) | 
| 155 | 154 | iuneq2dv 5015 | . . . . . . . . . . . . 13
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪
𝑦 ∈ 𝐵 (𝐹 “ (◡𝐹 “ {𝑦})) = ∪
𝑦 ∈ 𝐵 {𝑦}) | 
| 156 |  | iunid 5059 | . . . . . . . . . . . . 13
⊢ ∪ 𝑦 ∈ 𝐵 {𝑦} = 𝐵 | 
| 157 | 155, 156 | eqtrdi 2792 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪
𝑦 ∈ 𝐵 (𝐹 “ (◡𝐹 “ {𝑦})) = 𝐵) | 
| 158 | 146, 157 | sseqtrd 4019 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪
𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝐹 “ 𝑧) ⊆ 𝐵) | 
| 159 | 158 | ad2antrr 726 | . . . . . . . . . 10
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ∪
𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝐹 “ 𝑧) ⊆ 𝐵) | 
| 160 | 142, 159 | eqsstrid 4021 | . . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝐹 “ ∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ⊆ 𝐵) | 
| 161 | 141, 160 | sstrd 3993 | . . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝐹 “ ran 𝑓) ⊆ 𝐵) | 
| 162 | 40 | adantr 480 | . . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 163 | 162 | ffund 6739 | . . . . . . . . . . . . 13
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → Fun 𝑓) | 
| 164 |  | simpr 484 | . . . . . . . . . . . . . . 15
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝐵) | 
| 165 | 53, 55 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ◡𝐹 ∈ V) | 
| 166 | 165 | ad3antrrr 730 | . . . . . . . . . . . . . . . 16
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → ◡𝐹 ∈ V) | 
| 167 |  | imaexg 7936 | . . . . . . . . . . . . . . . 16
⊢ (◡𝐹 ∈ V → (◡𝐹 “ {𝑡}) ∈ V) | 
| 168 | 166, 167 | syl 17 | . . . . . . . . . . . . . . 15
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → (◡𝐹 “ {𝑡}) ∈ V) | 
| 169 | 1, 129 | elrnmpt1s 5969 | . . . . . . . . . . . . . . 15
⊢ ((𝑡 ∈ 𝐵 ∧ (◡𝐹 “ {𝑡}) ∈ V) → (◡𝐹 “ {𝑡}) ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 170 | 164, 168,
169 | syl2anc 584 | . . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → (◡𝐹 “ {𝑡}) ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 171 | 162 | fdmd 6745 | . . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → dom 𝑓 = ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) | 
| 172 | 170, 171 | eleqtrrd 2843 | . . . . . . . . . . . . 13
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → (◡𝐹 “ {𝑡}) ∈ dom 𝑓) | 
| 173 |  | fvelrn 7095 | . . . . . . . . . . . . 13
⊢ ((Fun
𝑓 ∧ (◡𝐹 “ {𝑡}) ∈ dom 𝑓) → (𝑓‘(◡𝐹 “ {𝑡})) ∈ ran 𝑓) | 
| 174 | 163, 172,
173 | syl2anc 584 | . . . . . . . . . . . 12
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → (𝑓‘(◡𝐹 “ {𝑡})) ∈ ran 𝑓) | 
| 175 | 15 | ad3antrrr 730 | . . . . . . . . . . . . 13
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → 𝐹 Fn 𝐴) | 
| 176 |  | simplr 768 | . . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) | 
| 177 |  | fveq2 6905 | . . . . . . . . . . . . . . . . 17
⊢ (𝑧 = (◡𝐹 “ {𝑡}) → (𝑓‘𝑧) = (𝑓‘(◡𝐹 “ {𝑡}))) | 
| 178 |  | id 22 | . . . . . . . . . . . . . . . . 17
⊢ (𝑧 = (◡𝐹 “ {𝑡}) → 𝑧 = (◡𝐹 “ {𝑡})) | 
| 179 | 177, 178 | eleq12d 2834 | . . . . . . . . . . . . . . . 16
⊢ (𝑧 = (◡𝐹 “ {𝑡}) → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡}))) | 
| 180 | 179 | rspcv 3617 | . . . . . . . . . . . . . . 15
⊢ ((◡𝐹 “ {𝑡}) ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧 → (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡}))) | 
| 181 | 180 | imp 406 | . . . . . . . . . . . . . 14
⊢ (((◡𝐹 “ {𝑡}) ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡})) | 
| 182 | 170, 176,
181 | syl2anc 584 | . . . . . . . . . . . . 13
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡})) | 
| 183 |  | fniniseg 7079 | . . . . . . . . . . . . . 14
⊢ (𝐹 Fn 𝐴 → ((𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡}) ↔ ((𝑓‘(◡𝐹 “ {𝑡})) ∈ 𝐴 ∧ (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡))) | 
| 184 | 183 | simplbda 499 | . . . . . . . . . . . . 13
⊢ ((𝐹 Fn 𝐴 ∧ (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡})) → (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡) | 
| 185 | 175, 182,
184 | syl2anc 584 | . . . . . . . . . . . 12
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡) | 
| 186 |  | fveqeq2 6914 | . . . . . . . . . . . . 13
⊢ (𝑘 = (𝑓‘(◡𝐹 “ {𝑡})) → ((𝐹‘𝑘) = 𝑡 ↔ (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡)) | 
| 187 | 186 | rspcev 3621 | . . . . . . . . . . . 12
⊢ (((𝑓‘(◡𝐹 “ {𝑡})) ∈ ran 𝑓 ∧ (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡) → ∃𝑘 ∈ ran 𝑓(𝐹‘𝑘) = 𝑡) | 
| 188 | 174, 185,
187 | syl2anc 584 | . . . . . . . . . . 11
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → ∃𝑘 ∈ ran 𝑓(𝐹‘𝑘) = 𝑡) | 
| 189 | 70 | adantr 480 | . . . . . . . . . . . 12
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → ran 𝑓 ⊆ 𝐴) | 
| 190 | 175, 189 | fvelimabd 6981 | . . . . . . . . . . 11
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → (𝑡 ∈ (𝐹 “ ran 𝑓) ↔ ∃𝑘 ∈ ran 𝑓(𝐹‘𝑘) = 𝑡)) | 
| 191 | 188, 190 | mpbird 257 | . . . . . . . . . 10
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ (𝐹 “ ran 𝑓)) | 
| 192 | 191 | ex 412 | . . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑡 ∈ 𝐵 → 𝑡 ∈ (𝐹 “ ran 𝑓))) | 
| 193 | 192 | ssrdv 3988 | . . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → 𝐵 ⊆ (𝐹 “ ran 𝑓)) | 
| 194 | 161, 193 | eqssd 4000 | . . . . . . 7
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝐹 “ ran 𝑓) = 𝐵) | 
| 195 | 138, 194 | jca 511 | . . . . . 6
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (ran 𝑓 ≈ 𝐵 ∧ (𝐹 “ ran 𝑓) = 𝐵)) | 
| 196 |  | breq1 5145 | . . . . . . . 8
⊢ (𝑥 = ran 𝑓 → (𝑥 ≈ 𝐵 ↔ ran 𝑓 ≈ 𝐵)) | 
| 197 |  | imaeq2 6073 | . . . . . . . . 9
⊢ (𝑥 = ran 𝑓 → (𝐹 “ 𝑥) = (𝐹 “ ran 𝑓)) | 
| 198 | 197 | eqeq1d 2738 | . . . . . . . 8
⊢ (𝑥 = ran 𝑓 → ((𝐹 “ 𝑥) = 𝐵 ↔ (𝐹 “ ran 𝑓) = 𝐵)) | 
| 199 | 196, 198 | anbi12d 632 | . . . . . . 7
⊢ (𝑥 = ran 𝑓 → ((𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵) ↔ (ran 𝑓 ≈ 𝐵 ∧ (𝐹 “ ran 𝑓) = 𝐵))) | 
| 200 | 199 | rspcev 3621 | . . . . . 6
⊢ ((ran
𝑓 ∈ 𝒫 𝐴 ∧ (ran 𝑓 ≈ 𝐵 ∧ (𝐹 “ ran 𝑓) = 𝐵)) → ∃𝑥 ∈ 𝒫 𝐴(𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵)) | 
| 201 | 71, 195, 200 | syl2anc 584 | . . . . 5
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ∃𝑥 ∈ 𝒫 𝐴(𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵)) | 
| 202 | 201 | anasss 466 | . . . 4
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ (𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧)) → ∃𝑥 ∈ 𝒫 𝐴(𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵)) | 
| 203 | 202 | ex 412 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ((𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ∃𝑥 ∈ 𝒫 𝐴(𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵))) | 
| 204 | 203 | exlimdv 1932 | . 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (∃𝑓(𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran
(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → ∃𝑥 ∈ 𝒫 𝐴(𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵))) | 
| 205 | 36, 204 | mpd 15 | 1
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∃𝑥 ∈ 𝒫 𝐴(𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵)) |