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Theorem fnpreimac 30172
Description: Choose a set 𝑥 containing a preimage of each element of a given set 𝐵. (Contributed by Thierry Arnoux, 7-May-2023.)
Assertion
Ref Expression
fnpreimac ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∃𝑥 ∈ 𝒫 𝐴(𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹
Allowed substitution hint:   𝑉(𝑥)
Proof of Theorem fnpreimac
Dummy variables 𝑓 𝑡 𝑢 𝑣 𝑦 𝑧 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2772 . . . . . . . . 9 (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) = (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))
21elrnmpt 5665 . . . . . . . 8 (𝑧 ∈ V → (𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ↔ ∃𝑦 ∈ 𝐵 𝑧 = (◡𝐹 “ {𝑦})))
32elv 3414 . . . . . . 7 (𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ↔ ∃𝑦 ∈ 𝐵 𝑧 = (◡𝐹 “ {𝑦}))
4 simpr 477 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (◡𝐹 “ {𝑦})) → 𝑧 = (◡𝐹 “ {𝑦}))
5 simpl3 1173 . . . . . . . . . . . 12 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → 𝐵 ⊆ ran 𝐹)
6 simpr 477 . . . . . . . . . . . 12 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
75, 6sseldd 3853 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ran 𝐹)
8 inisegn0 5795 . . . . . . . . . . 11 (𝑦 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑦}) ≠ ∅)
97, 8sylib 210 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → (◡𝐹 “ {𝑦}) ≠ ∅)
109adantr 473 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (◡𝐹 “ {𝑦})) → (◡𝐹 “ {𝑦}) ≠ ∅)
114, 10eqnetrd 3028 . . . . . . . 8 ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (◡𝐹 “ {𝑦})) → 𝑧 ≠ ∅)
1211r19.29an 3227 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ ∃𝑦 ∈ 𝐵 𝑧 = (◡𝐹 “ {𝑦})) → 𝑧 ≠ ∅)
133, 12sylan2b 584 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) → 𝑧 ≠ ∅)
1413ralrimiva 3126 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))𝑧 ≠ ∅)
15 simp2 1117 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐹 Fn 𝐴)
16 simp1 1116 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐴 ∈ 𝑉)
1715, 16jca 504 . . . . . . . . . 10 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉))
18 fnex 6800 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
19 rnexg 7423 . . . . . . . . . 10 (𝐹 ∈ V → ran 𝐹 ∈ V)
2017, 18, 193syl 18 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ran 𝐹 ∈ V)
21 simp3 1118 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐵 ⊆ ran 𝐹)
2220, 21ssexd 5078 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐵 ∈ V)
23 mptexg 6804 . . . . . . . 8 (𝐵 ∈ V → (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V)
24 rnexg 7423 . . . . . . . 8 ((𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V → ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V)
2522, 23, 243syl 18 . . . . . . 7 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V)
26 fvi 6562 . . . . . . 7 (ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V → ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) = ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))
2725, 26syl 17 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) = ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))
2827raleqdv 3349 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (∀𝑧 ∈ ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))𝑧 ≠ ∅ ↔ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))𝑧 ≠ ∅))
2914, 28mpbird 249 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∀𝑧 ∈ ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))𝑧 ≠ ∅)
30 fvex 6506 . . . . 5 ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∈ V
3130ac5b 9692 . . . 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))
3229, 31syl 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))
3327unieqd 4716 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪ ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) = ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))
3427, 33feq23d 6333 . . . . 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}))))
3527raleqdv 3349 . . . . 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))
3634, 35anbi12d 621 . . . 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)))
3736exbidv 1880 . . 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)))
3832, 37mpbid 224 . 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))
39 vex 3412 . . . . . . . . 9 𝑓 ∈ V
4039rnex 7426 . . . . . . . 8 ran 𝑓 ∈ V
4140a1i 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)
42 simplr 756 . . . . . . . . 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})))
43 frn 6344 . . . . . . . . 9 (𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → ran 𝑓 ⊆ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))
4442, 43syl 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})))
45 nfv 1873 . . . . . . . . . . . . 13 Ⅎ𝑦(𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹)
46 nfcv 2926 . . . . . . . . . . . . . 14 Ⅎ𝑦𝑓
47 nfmpt1 5019 . . . . . . . . . . . . . . 15 Ⅎ𝑦(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))
4847nfrn 5661 . . . . . . . . . . . . . 14 Ⅎ𝑦ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))
4948nfuni 4712 . . . . . . . . . . . . . 14 Ⅎ𝑦∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))
5046, 48, 49nff 6334 . . . . . . . . . . . . 13 Ⅎ𝑦 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))
5145, 50nfan 1862 . . . . . . . . . . . 12 Ⅎ𝑦((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))
52 nfv 1873 . . . . . . . . . . . . 13 Ⅎ𝑦(𝑓‘𝑧) ∈ 𝑧
5348, 52nfral 3168 . . . . . . . . . . . 12 Ⅎ𝑦∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧
5451, 53nfan 1862 . . . . . . . . . . 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)
5517, 18syl 17 . . . . . . . . . . . . . . 15 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐹 ∈ V)
5655ad3antrrr 717 . . . . . . . . . . . . . 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)
57 cnvexg 7438 . . . . . . . . . . . . . 14 (𝐹 ∈ V → ◡𝐹 ∈ V)
58 imaexg 7429 . . . . . . . . . . . . . 14 (◡𝐹 ∈ V → (◡𝐹 “ {𝑦}) ∈ V)
5956, 57, 583syl 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)
60 cnvimass 5783 . . . . . . . . . . . . . . 15 (◡𝐹 “ {𝑦}) ⊆ dom 𝐹
6160a1i 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)
62 fndm 6282 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6315, 62syl 17 . . . . . . . . . . . . . . 15 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → dom 𝐹 = 𝐴)
6463ad3antrrr 717 . . . . . . . . . . . . . 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)
6561, 64sseqtrd 3891 . . . . . . . . . . . . 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)
6659, 65elpwd 4425 . . . . . . . . . . . 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)
6766ex 405 . . . . . . . . . . 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))
6854, 67ralrimi 3160 . . . . . . . . . 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)
691rnmptss 6703 . . . . . . . . . 10 (∀𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦}) ∈ 𝒫 𝐴 → ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ⊆ 𝒫 𝐴)
7068, 69syl 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)
71 sspwuni 4882 . . . . . . . . 9 (ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ⊆ 𝒫 𝐴 ↔ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ⊆ 𝐴)
7270, 71sylib 210 . . . . . . . 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)
7344, 72sstrd 3862 . . . . . . 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)
7441, 73elpwd 4425 . . . . . 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)
75 fnfun 6280 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn 𝐴 → Fun 𝐹)
7615, 75syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → Fun 𝐹)
7776ad5antr 721 . . . . . . . . . . . . . . . . . . 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)
78 sndisj 4915 . . . . . . . . . . . . . . . . . . 19 Disj 𝑦 ∈ 𝐵 {𝑦}
79 disjpreima 30094 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹 ∧ Disj 𝑦 ∈ 𝐵 {𝑦}) → Disj 𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦}))
8077, 78, 79sylancl 577 . . . . . . . . . . . . . . . . . 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}))
81 disjrnmpt 30095 . . . . . . . . . . . . . . . . . 18 (Disj 𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦}) → Disj 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))𝑧)
8280, 81syl 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)
83 simpllr 763 . . . . . . . . . . . . . . . . 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})))
84 simplr 756 . . . . . . . . . . . . . . . . 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})))
85 simp-4r 771 . . . . . . . . . . . . . . . . . 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)
86 fveq2 6493 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑢 → (𝑓‘𝑧) = (𝑓‘𝑢))
87 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑢 → 𝑧 = 𝑢)
8886, 87eleq12d 2854 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝑢 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑢))
8988rspcv 3525 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧 → (𝑓‘𝑢) ∈ 𝑢))
9089imp 398 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑓‘𝑢) ∈ 𝑢)
9183, 85, 90syl2anc 576 . . . . . . . . . . . . . . . . 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)
92 simpr 477 . . . . . . . . . . . . . . . . . 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))
93 fveq2 6493 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝑣 → (𝑓‘𝑧) = (𝑓‘𝑣))
94 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝑣 → 𝑧 = 𝑣)
9593, 94eleq12d 2854 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑣 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑣) ∈ 𝑣))
9695rspcv 3525 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧 → (𝑓‘𝑣) ∈ 𝑣))
9796imp 398 . . . . . . . . . . . . . . . . . . 19 ((𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑓‘𝑣) ∈ 𝑣)
9884, 85, 97syl2anc 576 . . . . . . . . . . . . . . . . . 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)
9992, 98eqeltrd 2860 . . . . . . . . . . . . . . . . 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)
10087, 94disji 4908 . . . . . . . . . . . . . . . . 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)
10182, 83, 84, 91, 99, 100syl122anc 1359 . . . . . . . . . . . . . . . 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)
102101ex 405 . . . . . . . . . . . . . . 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))
103102anasss 459 . . . . . . . . . . . . . 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))
104103ralrimivva 3135 . . . . . . . . . . . . 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))
10542, 104jca 504 . . . . . . . . . . . 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)))
106 dff13 6832 . . . . . . . . . . . 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)))
107105, 106sylibr 226 . . . . . . . . . . 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})))
108 f1f1orn 6449 . . . . . . . . . . 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)
109107, 108syl 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)
110 f1oen3g 8316 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))–1-1-onto→ran 𝑓) → ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ≈ ran 𝑓)
11139, 109, 110sylancr 578 . . . . . . . . 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)
112111ensymd 8351 . . . . . . . 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})))
11322, 23syl 17 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V)
114113ad2antrr 713 . . . . . . . . . 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)
11559ex 405 . . . . . . . . . . . . . 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))
11654, 115ralrimi 3160 . . . . . . . . . . . . 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)
11776ad5antr 721 . . . . . . . . . . . . . . . . . . 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)
118 simpr 477 . . . . . . . . . . . . . . . . . . 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)
11921ad5antr 721 . . . . . . . . . . . . . . . . . . . 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)
120 simpllr 763 . . . . . . . . . . . . . . . . . . . 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)
121119, 120sseldd 3853 . . . . . . . . . . . . . . . . . . 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)
122 simplr 756 . . . . . . . . . . . . . . . . . . . 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)
123119, 122sseldd 3853 . . . . . . . . . . . . . . . . . . 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)
124117, 118, 121, 123preimane 30171 . . . . . . . . . . . . . . . . . 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}))
125124ex 405 . . . . . . . . . . . . . . . . 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})))
126125necon4d 2985 . . . . . . . . . . . . . . . 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))
127126ralrimiva 3126 . . . . . . . . . . . . . . 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))
128127ex 405 . . . . . . . . . . . . . 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)))
12954, 128ralrimi 3160 . . . . . . . . . . . . 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))
130116, 129jca 504 . . . . . . . . . . . 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)))
131 sneq 4445 . . . . . . . . . . . . . 14 (𝑦 = 𝑡 → {𝑦} = {𝑡})
132131imaeq2d 5764 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑡}))
1331, 132f1mpt 6838 . . . . . . . . . . . 12 ((𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})):𝐵–1-1→V ↔ (∀𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦}) ∈ V ∧ ∀𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐵 ((◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑡}) → 𝑦 = 𝑡)))
134130, 133sylibr 226 . . . . . . . . . . 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)
135 f1f1orn 6449 . . . . . . . . . . 11 ((𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})):𝐵–1-1→V → (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})):𝐵–1-1-onto→ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))
136134, 135syl 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})))
137 f1oen3g 8316 . . . . . . . . . 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})))
138114, 136, 137syl2anc 576 . . . . . . . . 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})))
139138ensymd 8351 . . . . . . . 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)
140 entr 8352 . . . . . . . 8 ((ran 𝑓 ≈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ≈ 𝐵) → ran 𝑓 ≈ 𝐵)
141112, 139, 140syl2anc 576 . . . . . . 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)
142 imass2 5799 . . . . . . . . . . 11 (ran 𝑓 ⊆ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (𝐹 “ ran 𝑓) ⊆ (𝐹 “ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))))
14343, 142syl 17 . . . . . . . . . 10 (𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (𝐹 “ ran 𝑓) ⊆ (𝐹 “ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))))
14442, 143syl 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}))))
145 imauni 6824 . . . . . . . . . 10 (𝐹 “ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) = ∪ 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝐹 “ 𝑧)
146 imaeq2 5760 . . . . . . . . . . . . 13 (𝑧 = (◡𝐹 “ {𝑦}) → (𝐹 “ 𝑧) = (𝐹 “ (◡𝐹 “ {𝑦})))
14755adantr 473 . . . . . . . . . . . . . 14 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → 𝐹 ∈ V)
148147, 57, 583syl 18 . . . . . . . . . . . . 13 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → (◡𝐹 “ {𝑦}) ∈ V)
149146, 148iunrnmptss 30080 . . . . . . . . . . . 12 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪ 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝐹 “ 𝑧) ⊆ ∪ 𝑦 ∈ 𝐵 (𝐹 “ (◡𝐹 “ {𝑦})))
150 funimacnv 6262 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 → (𝐹 “ (◡𝐹 “ {𝑦})) = ({𝑦} ∩ ran 𝐹))
15176, 150syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (𝐹 “ (◡𝐹 “ {𝑦})) = ({𝑦} ∩ ran 𝐹))
152151adantr 473 . . . . . . . . . . . . . . 15 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → (𝐹 “ (◡𝐹 “ {𝑦})) = ({𝑦} ∩ ran 𝐹))
1536snssd 4610 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → {𝑦} ⊆ 𝐵)
154153, 5sstrd 3862 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → {𝑦} ⊆ ran 𝐹)
155 df-ss 3837 . . . . . . . . . . . . . . . 16 ({𝑦} ⊆ ran 𝐹 ↔ ({𝑦} ∩ ran 𝐹) = {𝑦})
156154, 155sylib 210 . . . . . . . . . . . . . . 15 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → ({𝑦} ∩ ran 𝐹) = {𝑦})
157152, 156eqtrd 2808 . . . . . . . . . . . . . 14 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → (𝐹 “ (◡𝐹 “ {𝑦})) = {𝑦})
158157iuneq2dv 4809 . . . . . . . . . . . . 13 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪ 𝑦 ∈ 𝐵 (𝐹 “ (◡𝐹 “ {𝑦})) = ∪ 𝑦 ∈ 𝐵 {𝑦})
159 iunid 4844 . . . . . . . . . . . . 13 ∪ 𝑦 ∈ 𝐵 {𝑦} = 𝐵
160158, 159syl6eq 2824 . . . . . . . . . . . 12 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪ 𝑦 ∈ 𝐵 (𝐹 “ (◡𝐹 “ {𝑦})) = 𝐵)
161149, 160sseqtrd 3891 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪ 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝐹 “ 𝑧) ⊆ 𝐵)
162161ad2antrr 713 . . . . . . . . . 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)
163145, 162syl5eqss 3899 . . . . . . . . 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)
164144, 163sstrd 3862 . . . . . . . 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)
16542adantr 473 . . . . . . . . . . . . . 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})))
166165ffund 6342 . . . . . . . . . . . . 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)
167 simpr 477 . . . . . . . . . . . . . . 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)
16855, 57syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ◡𝐹 ∈ V)
169168ad3antrrr 717 . . . . . . . . . . . . . . . 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)
170 imaexg 7429 . . . . . . . . . . . . . . . 16 (◡𝐹 ∈ V → (◡𝐹 “ {𝑡}) ∈ V)
171169, 170syl 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)
1721, 132elrnmpt1s 5666 . . . . . . . . . . . . . . 15 ((𝑡 ∈ 𝐵 ∧ (◡𝐹 “ {𝑡}) ∈ V) → (◡𝐹 “ {𝑡}) ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))
173167, 171, 172syl2anc 576 . . . . . . . . . . . . . 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})))
174165fdmd 6347 . . . . . . . . . . . . . 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})))
175173, 174eleqtrrd 2863 . . . . . . . . . . . . 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)
176 fvelrn 6663 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ (◡𝐹 “ {𝑡}) ∈ dom 𝑓) → (𝑓‘(◡𝐹 “ {𝑡})) ∈ ran 𝑓)
177166, 175, 176syl2anc 576 . . . . . . . . . . . 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)
17815ad3antrrr 717 . . . . . . . . . . . . 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)
179 simplr 756 . . . . . . . . . . . . . 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)
180 fveq2 6493 . . . . . . . . . . . . . . . . 17 (𝑧 = (◡𝐹 “ {𝑡}) → (𝑓‘𝑧) = (𝑓‘(◡𝐹 “ {𝑡})))
181 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = (◡𝐹 “ {𝑡}) → 𝑧 = (◡𝐹 “ {𝑡}))
182180, 181eleq12d 2854 . . . . . . . . . . . . . . . 16 (𝑧 = (◡𝐹 “ {𝑡}) → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡})))
183182rspcv 3525 . . . . . . . . . . . . . . 15 ((◡𝐹 “ {𝑡}) ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧 → (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡})))
184183imp 398 . . . . . . . . . . . . . 14 (((◡𝐹 “ {𝑡}) ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡}))
185173, 179, 184syl2anc 576 . . . . . . . . . . . . 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}))
186 fniniseg 6649 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → ((𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡}) ↔ ((𝑓‘(◡𝐹 “ {𝑡})) ∈ 𝐴 ∧ (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡)))
187186simplbda 492 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴 ∧ (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡})) → (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡)
188178, 185, 187syl2anc 576 . . . . . . . . . . . 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)
189 fveqeq2 6502 . . . . . . . . . . . . 13 (𝑘 = (𝑓‘(◡𝐹 “ {𝑡})) → ((𝐹‘𝑘) = 𝑡 ↔ (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡))
190189rspcev 3529 . . . . . . . . . . . 12 (((𝑓‘(◡𝐹 “ {𝑡})) ∈ ran 𝑓 ∧ (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡) → ∃𝑘 ∈ ran 𝑓(𝐹‘𝑘) = 𝑡)
191177, 188, 190syl2anc 576 . . . . . . . . . . 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)
19273adantr 473 . . . . . . . . . . . 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)
193178, 192fvelimabd 6561 . . . . . . . . . . 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))
194191, 193mpbird 249 . . . . . . . . . 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))
195194ex 405 . . . . . . . . 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)))
196195ssrdv 3858 . . . . . . . 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))
197164, 196eqssd 3869 . . . . . . 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)
198141, 197jca 504 . . . . . 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))
199 breq1 4926 . . . . . . . 8 (𝑥 = ran 𝑓 → (𝑥 ≈ 𝐵 ↔ ran 𝑓 ≈ 𝐵))
200 imaeq2 5760 . . . . . . . . 9 (𝑥 = ran 𝑓 → (𝐹 “ 𝑥) = (𝐹 “ ran 𝑓))
201200eqeq1d 2774 . . . . . . . 8 (𝑥 = ran 𝑓 → ((𝐹 “ 𝑥) = 𝐵 ↔ (𝐹 “ ran 𝑓) = 𝐵))
202199, 201anbi12d 621 . . . . . . 7 (𝑥 = ran 𝑓 → ((𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵) ↔ (ran 𝑓 ≈ 𝐵 ∧ (𝐹 “ ran 𝑓) = 𝐵)))
203202rspcev 3529 . . . . . 6 ((ran 𝑓 ∈ 𝒫 𝐴 ∧ (ran 𝑓 ≈ 𝐵 ∧ (𝐹 “ ran 𝑓) = 𝐵)) → ∃𝑥 ∈ 𝒫 𝐴(𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵))
20474, 198, 203syl2anc 576 . . . . 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))
205204anasss 459 . . . 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))
206205ex 405 . . 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)))
207206exlimdv 1892 . 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)))
20838, 207mpd 15 1 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∃𝑥 ∈ 𝒫 𝐴(𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507  ∃wex 1742   ∈ wcel 2050   ≠ wne 2961  ∀wral 3082  ∃wrex 3083  Vcvv 3409   ∩ cin 3822   ⊆ wss 3823  ∅c0 4172  𝒫 cpw 4416  {csn 4435  ∪ cuni 4706  ∪ ciun 4786  Disj wdisj 4891   class class class wbr 4923   ↦ cmpt 5002   I cid 5305  ◡ccnv 5400  dom cdm 5401  ran crn 5402   “ cima 5404  Fun wfun 6176   Fn wfn 6177  ⟶wf 6178  –1-1→wf1 6179  –1-1-onto→wf1o 6181  ‘cfv 6182   ≈ cen 8297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-ac2 9677
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-disj 4892  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-se 5361  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-wrecs 7744  df-recs 7806  df-er 8083  df-en 8301  df-card 9156  df-ac 9330
This theorem is referenced by: (None)
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