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Theorem fnpreimac 30910
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 2738 . . . . . . . . 9 (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) = (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))
21elrnmpt 5854 . . . . . . . 8 (𝑧 ∈ V → (𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ↔ ∃𝑦 ∈ 𝐵 𝑧 = (◡𝐹 “ {𝑦})))
32elv 3428 . . . . . . 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 1191 . . . . . . . . . . . 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)
75, 6sseldd 3918 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ran 𝐹)
8 inisegn0 5995 . . . . . . . . . . 11 (𝑦 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑦}) ≠ ∅)
97, 8sylib 217 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → (◡𝐹 “ {𝑦}) ≠ ∅)
109adantr 480 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (◡𝐹 “ {𝑦})) → (◡𝐹 “ {𝑦}) ≠ ∅)
114, 10eqnetrd 3010 . . . . . . . 8 ((((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (◡𝐹 “ {𝑦})) → 𝑧 ≠ ∅)
1211r19.29an 3216 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ ∃𝑦 ∈ 𝐵 𝑧 = (◡𝐹 “ {𝑦})) → 𝑧 ≠ ∅)
133, 12sylan2b 593 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) → 𝑧 ≠ ∅)
1413ralrimiva 3107 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))𝑧 ≠ ∅)
15 simp2 1135 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐹 Fn 𝐴)
16 simp1 1134 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐴 ∈ 𝑉)
1715, 16jca 511 . . . . . . . . . 10 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉))
18 fnex 7075 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
19 rnexg 7725 . . . . . . . . . 10 (𝐹 ∈ V → ran 𝐹 ∈ V)
2017, 18, 193syl 18 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ran 𝐹 ∈ V)
21 simp3 1136 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐵 ⊆ ran 𝐹)
2220, 21ssexd 5243 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → 𝐵 ∈ V)
23 mptexg 7079 . . . . . . . 8 (𝐵 ∈ V → (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V)
24 rnexg 7725 . . . . . . . 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 6826 . . . . . . 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 3339 . . . . 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 256 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∀𝑧 ∈ ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))𝑧 ≠ ∅)
30 fvex 6769 . . . . 5 ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) ∈ V
3130ac5b 10165 . . . 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 4850 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪ ( I ‘ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) = ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))
3427, 33feq23d 6579 . . . . 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 3339 . . . . 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 630 . . . 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 1925 . . 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 231 . 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 3426 . . . . . . . . 9 𝑓 ∈ V
4039rnex 7733 . . . . . . . 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 765 . . . . . . . . 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 6591 . . . . . . . . 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 1918 . . . . . . . . . . . . 13 Ⅎ𝑦(𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹)
46 nfcv 2906 . . . . . . . . . . . . . 14 Ⅎ𝑦𝑓
47 nfmpt1 5178 . . . . . . . . . . . . . . 15 Ⅎ𝑦(𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))
4847nfrn 5850 . . . . . . . . . . . . . 14 Ⅎ𝑦ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))
4948nfuni 4843 . . . . . . . . . . . . . 14 Ⅎ𝑦∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))
5046, 48, 49nff 6580 . . . . . . . . . . . . 13 Ⅎ𝑦 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))
5145, 50nfan 1903 . . . . . . . . . . . 12 Ⅎ𝑦((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))
52 nfv 1918 . . . . . . . . . . . . 13 Ⅎ𝑦(𝑓‘𝑧) ∈ 𝑧
5348, 52nfralw 3149 . . . . . . . . . . . 12 Ⅎ𝑦∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧
5451, 53nfan 1903 . . . . . . . . . . 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 726 . . . . . . . . . . . . . 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 7745 . . . . . . . . . . . . . 14 (𝐹 ∈ V → ◡𝐹 ∈ V)
58 imaexg 7736 . . . . . . . . . . . . . 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 5978 . . . . . . . . . . . . . . 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)
6215fndmd 6522 . . . . . . . . . . . . . . 15 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → dom 𝐹 = 𝐴)
6362ad3antrrr 726 . . . . . . . . . . . . . 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)
6461, 63sseqtrd 3957 . . . . . . . . . . . . 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)
6559, 64elpwd 4538 . . . . . . . . . . . 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)
6665ex 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))
6754, 66ralrimi 3139 . . . . . . . . . 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)
681rnmptss 6978 . . . . . . . . . 10 (∀𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦}) ∈ 𝒫 𝐴 → ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ⊆ 𝒫 𝐴)
6967, 68syl 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)
70 sspwuni 5025 . . . . . . . . 9 (ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ⊆ 𝒫 𝐴 ↔ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ⊆ 𝐴)
7169, 70sylib 217 . . . . . . . 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)
7244, 71sstrd 3927 . . . . . . 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)
7341, 72elpwd 4538 . . . . . 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)
74 fnfun 6517 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn 𝐴 → Fun 𝐹)
7515, 74syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → Fun 𝐹)
7675ad5antr 730 . . . . . . . . . . . . . . . . . . 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)
77 sndisj 5061 . . . . . . . . . . . . . . . . . . 19 Disj 𝑦 ∈ 𝐵 {𝑦}
78 disjpreima 30824 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹 ∧ Disj 𝑦 ∈ 𝐵 {𝑦}) → Disj 𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦}))
7976, 77, 78sylancl 585 . . . . . . . . . . . . . . . . . 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}))
80 disjrnmpt 30825 . . . . . . . . . . . . . . . . . 18 (Disj 𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦}) → Disj 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))𝑧)
8179, 80syl 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)
82 simpllr 772 . . . . . . . . . . . . . . . . 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})))
83 simplr 765 . . . . . . . . . . . . . . . . 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})))
84 simp-4r 780 . . . . . . . . . . . . . . . . . 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)
85 fveq2 6756 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑢 → (𝑓‘𝑧) = (𝑓‘𝑢))
86 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑢 → 𝑧 = 𝑢)
8785, 86eleq12d 2833 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝑢 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑢))
8887rspcv 3547 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧 → (𝑓‘𝑢) ∈ 𝑢))
8988imp 406 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑓‘𝑢) ∈ 𝑢)
9082, 84, 89syl2anc 583 . . . . . . . . . . . . . . . . 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)
91 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))
92 fveq2 6756 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝑣 → (𝑓‘𝑧) = (𝑓‘𝑣))
93 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝑣 → 𝑧 = 𝑣)
9492, 93eleq12d 2833 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑣 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑣) ∈ 𝑣))
9594rspcv 3547 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧 → (𝑓‘𝑣) ∈ 𝑣))
9695imp 406 . . . . . . . . . . . . . . . . . . 19 ((𝑣 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑓‘𝑣) ∈ 𝑣)
9783, 84, 96syl2anc 583 . . . . . . . . . . . . . . . . . 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)
9891, 97eqeltrd 2839 . . . . . . . . . . . . . . . . 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)
9986, 93disji 5053 . . . . . . . . . . . . . . . . 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)
10081, 82, 83, 90, 98, 99syl122anc 1377 . . . . . . . . . . . . . . . 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)
101100ex 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))
102101anasss 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))
103102ralrimivva 3114 . . . . . . . . . . . . 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))
10442, 103jca 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)))
105 dff13 7109 . . . . . . . . . . . 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)))
106104, 105sylibr 233 . . . . . . . . . . 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})))
107 f1f1orn 6711 . . . . . . . . . . 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)
108106, 107syl 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)
109 f1oen3g 8709 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))–1-1-onto→ran 𝑓) → ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ≈ ran 𝑓)
11039, 108, 109sylancr 586 . . . . . . . . 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)
111110ensymd 8746 . . . . . . . 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})))
11222, 23syl 17 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∈ V)
113112ad2antrr 722 . . . . . . . . . 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)
11459ex 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))
11554, 114ralrimi 3139 . . . . . . . . . . . . 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)
11675ad5antr 730 . . . . . . . . . . . . . . . . . . 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)
117 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)
11821ad5antr 730 . . . . . . . . . . . . . . . . . . . 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)
119 simpllr 772 . . . . . . . . . . . . . . . . . . . 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)
120118, 119sseldd 3918 . . . . . . . . . . . . . . . . . . 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)
121 simplr 765 . . . . . . . . . . . . . . . . . . . 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)
122118, 121sseldd 3918 . . . . . . . . . . . . . . . . . . 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)
123116, 117, 120, 122preimane 30909 . . . . . . . . . . . . . . . . . 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}))
124123ex 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})))
125124necon4d 2966 . . . . . . . . . . . . . . . 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))
126125ralrimiva 3107 . . . . . . . . . . . . . . 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))
127126ex 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)))
12854, 127ralrimi 3139 . . . . . . . . . . . . 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))
129115, 128jca 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)))
130 sneq 4568 . . . . . . . . . . . . . 14 (𝑦 = 𝑡 → {𝑦} = {𝑡})
131130imaeq2d 5958 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑡}))
1321, 131f1mpt 7115 . . . . . . . . . . . 12 ((𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})):𝐵–1-1→V ↔ (∀𝑦 ∈ 𝐵 (◡𝐹 “ {𝑦}) ∈ V ∧ ∀𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐵 ((◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑡}) → 𝑦 = 𝑡)))
133129, 132sylibr 233 . . . . . . . . . . 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)
134 f1f1orn 6711 . . . . . . . . . . 11 ((𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})):𝐵–1-1→V → (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})):𝐵–1-1-onto→ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))
135133, 134syl 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})))
136 f1oen3g 8709 . . . . . . . . . 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})))
137113, 135, 136syl2anc 583 . . . . . . . . 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})))
138137ensymd 8746 . . . . . . . 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)
139 entr 8747 . . . . . . . 8 ((ran 𝑓 ≈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ≈ 𝐵) → ran 𝑓 ≈ 𝐵)
140111, 138, 139syl2anc 583 . . . . . . 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)
141 imass2 5999 . . . . . . . . . . 11 (ran 𝑓 ⊆ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (𝐹 “ ran 𝑓) ⊆ (𝐹 “ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))))
14243, 141syl 17 . . . . . . . . . 10 (𝑓:ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))⟶∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (𝐹 “ ran 𝑓) ⊆ (𝐹 “ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))))
14342, 142syl 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}))))
144 imauni 7101 . . . . . . . . . 10 (𝐹 “ ∪ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))) = ∪ 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝐹 “ 𝑧)
145 imaeq2 5954 . . . . . . . . . . . . 13 (𝑧 = (◡𝐹 “ {𝑦}) → (𝐹 “ 𝑧) = (𝐹 “ (◡𝐹 “ {𝑦})))
14655adantr 480 . . . . . . . . . . . . . 14 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → 𝐹 ∈ V)
147146, 57, 583syl 18 . . . . . . . . . . . . 13 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → (◡𝐹 “ {𝑦}) ∈ V)
148145, 147iunrnmptss 30806 . . . . . . . . . . . 12 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪ 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝐹 “ 𝑧) ⊆ ∪ 𝑦 ∈ 𝐵 (𝐹 “ (◡𝐹 “ {𝑦})))
149 funimacnv 6499 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 → (𝐹 “ (◡𝐹 “ {𝑦})) = ({𝑦} ∩ ran 𝐹))
15075, 149syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → (𝐹 “ (◡𝐹 “ {𝑦})) = ({𝑦} ∩ ran 𝐹))
151150adantr 480 . . . . . . . . . . . . . . 15 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → (𝐹 “ (◡𝐹 “ {𝑦})) = ({𝑦} ∩ ran 𝐹))
1526snssd 4739 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → {𝑦} ⊆ 𝐵)
153152, 5sstrd 3927 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → {𝑦} ⊆ ran 𝐹)
154 df-ss 3900 . . . . . . . . . . . . . . . 16 ({𝑦} ⊆ ran 𝐹 ↔ ({𝑦} ∩ ran 𝐹) = {𝑦})
155153, 154sylib 217 . . . . . . . . . . . . . . 15 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → ({𝑦} ∩ ran 𝐹) = {𝑦})
156151, 155eqtrd 2778 . . . . . . . . . . . . . 14 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) ∧ 𝑦 ∈ 𝐵) → (𝐹 “ (◡𝐹 “ {𝑦})) = {𝑦})
157156iuneq2dv 4945 . . . . . . . . . . . . 13 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪ 𝑦 ∈ 𝐵 (𝐹 “ (◡𝐹 “ {𝑦})) = ∪ 𝑦 ∈ 𝐵 {𝑦})
158 iunid 4986 . . . . . . . . . . . . 13 ∪ 𝑦 ∈ 𝐵 {𝑦} = 𝐵
159157, 158eqtrdi 2795 . . . . . . . . . . . 12 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪ 𝑦 ∈ 𝐵 (𝐹 “ (◡𝐹 “ {𝑦})) = 𝐵)
160148, 159sseqtrd 3957 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∪ 𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝐹 “ 𝑧) ⊆ 𝐵)
161160ad2antrr 722 . . . . . . . . . 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)
162144, 161eqsstrid 3965 . . . . . . . . 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)
163143, 162sstrd 3927 . . . . . . . 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)
16442adantr 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})))
165164ffund 6588 . . . . . . . . . . . . 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)
166 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)
16755, 57syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ◡𝐹 ∈ V)
168167ad3antrrr 726 . . . . . . . . . . . . . . . 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)
169 imaexg 7736 . . . . . . . . . . . . . . . 16 (◡𝐹 ∈ V → (◡𝐹 “ {𝑡}) ∈ V)
170168, 169syl 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)
1711, 131elrnmpt1s 5855 . . . . . . . . . . . . . . 15 ((𝑡 ∈ 𝐵 ∧ (◡𝐹 “ {𝑡}) ∈ V) → (◡𝐹 “ {𝑡}) ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})))
172166, 170, 171syl2anc 583 . . . . . . . . . . . . . 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})))
173164fdmd 6595 . . . . . . . . . . . . . 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})))
174172, 173eleqtrrd 2842 . . . . . . . . . . . . 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)
175 fvelrn 6936 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ (◡𝐹 “ {𝑡}) ∈ dom 𝑓) → (𝑓‘(◡𝐹 “ {𝑡})) ∈ ran 𝑓)
176165, 174, 175syl2anc 583 . . . . . . . . . . . 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)
17715ad3antrrr 726 . . . . . . . . . . . . 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)
178 simplr 765 . . . . . . . . . . . . . 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)
179 fveq2 6756 . . . . . . . . . . . . . . . . 17 (𝑧 = (◡𝐹 “ {𝑡}) → (𝑓‘𝑧) = (𝑓‘(◡𝐹 “ {𝑡})))
180 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = (◡𝐹 “ {𝑡}) → 𝑧 = (◡𝐹 “ {𝑡}))
181179, 180eleq12d 2833 . . . . . . . . . . . . . . . 16 (𝑧 = (◡𝐹 “ {𝑡}) → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡})))
182181rspcv 3547 . . . . . . . . . . . . . . 15 ((◡𝐹 “ {𝑡}) ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) → (∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧 → (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡})))
183182imp 406 . . . . . . . . . . . . . 14 (((◡𝐹 “ {𝑡}) ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦 ∈ 𝐵 ↦ (◡𝐹 “ {𝑦}))(𝑓‘𝑧) ∈ 𝑧) → (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡}))
184172, 178, 183syl2anc 583 . . . . . . . . . . . . 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}))
185 fniniseg 6919 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → ((𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡}) ↔ ((𝑓‘(◡𝐹 “ {𝑡})) ∈ 𝐴 ∧ (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡)))
186185simplbda 499 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴 ∧ (𝑓‘(◡𝐹 “ {𝑡})) ∈ (◡𝐹 “ {𝑡})) → (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡)
187177, 184, 186syl2anc 583 . . . . . . . . . . . 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)
188 fveqeq2 6765 . . . . . . . . . . . . 13 (𝑘 = (𝑓‘(◡𝐹 “ {𝑡})) → ((𝐹‘𝑘) = 𝑡 ↔ (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡))
189188rspcev 3552 . . . . . . . . . . . 12 (((𝑓‘(◡𝐹 “ {𝑡})) ∈ ran 𝑓 ∧ (𝐹‘(𝑓‘(◡𝐹 “ {𝑡}))) = 𝑡) → ∃𝑘 ∈ ran 𝑓(𝐹‘𝑘) = 𝑡)
190176, 187, 189syl2anc 583 . . . . . . . . . . 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)
19172adantr 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)
192177, 191fvelimabd 6824 . . . . . . . . . . 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))
193190, 192mpbird 256 . . . . . . . . . 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))
194193ex 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)))
195194ssrdv 3923 . . . . . . . 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))
196163, 195eqssd 3934 . . . . . . 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)
197140, 196jca 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))
198 breq1 5073 . . . . . . . 8 (𝑥 = ran 𝑓 → (𝑥 ≈ 𝐵 ↔ ran 𝑓 ≈ 𝐵))
199 imaeq2 5954 . . . . . . . . 9 (𝑥 = ran 𝑓 → (𝐹 “ 𝑥) = (𝐹 “ ran 𝑓))
200199eqeq1d 2740 . . . . . . . 8 (𝑥 = ran 𝑓 → ((𝐹 “ 𝑥) = 𝐵 ↔ (𝐹 “ ran 𝑓) = 𝐵))
201198, 200anbi12d 630 . . . . . . 7 (𝑥 = ran 𝑓 → ((𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵) ↔ (ran 𝑓 ≈ 𝐵 ∧ (𝐹 “ ran 𝑓) = 𝐵)))
202201rspcev 3552 . . . . . 6 ((ran 𝑓 ∈ 𝒫 𝐴 ∧ (ran 𝑓 ≈ 𝐵 ∧ (𝐹 “ ran 𝑓) = 𝐵)) → ∃𝑥 ∈ 𝒫 𝐴(𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵))
20373, 197, 202syl2anc 583 . . . . 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))
204203anasss 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))
205204ex 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)))
206205exlimdv 1937 . 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)))
20738, 206mpd 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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836  ∪ ciun 4921  Disj wdisj 5035   class class class wbr 5070   ↦ cmpt 5153   I cid 5479  ◡ccnv 5579  dom cdm 5580  ran crn 5581   “ cima 5583  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  –1-1→wf1 6415  –1-1-onto→wf1o 6417  ‘cfv 6418   ≈ cen 8688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-ac2 10150
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-disj 5036  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-er 8456  df-en 8692  df-card 9628  df-ac 9803
This theorem is referenced by: (None)
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