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Theorem fnpreimac 30427
 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 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ∃𝑥 ∈ 𝒫 𝐴(𝑥𝐵 ∧ (𝐹𝑥) = 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 2824 . . . . . . . . 9 (𝑦𝐵 ↦ (𝐹 “ {𝑦})) = (𝑦𝐵 ↦ (𝐹 “ {𝑦}))
21elrnmpt 5815 . . . . . . . 8 (𝑧 ∈ V → (𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ↔ ∃𝑦𝐵 𝑧 = (𝐹 “ {𝑦})))
32elv 3485 . . . . . . 7 (𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ↔ ∃𝑦𝐵 𝑧 = (𝐹 “ {𝑦}))
4 simpr 488 . . . . . . . . 9 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) ∧ 𝑧 = (𝐹 “ {𝑦})) → 𝑧 = (𝐹 “ {𝑦}))
5 simpl3 1190 . . . . . . . . . . . 12 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) → 𝐵 ⊆ ran 𝐹)
6 simpr 488 . . . . . . . . . . . 12 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) → 𝑦𝐵)
75, 6sseldd 3954 . . . . . . . . . . 11 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) → 𝑦 ∈ ran 𝐹)
8 inisegn0 5948 . . . . . . . . . . 11 (𝑦 ∈ ran 𝐹 ↔ (𝐹 “ {𝑦}) ≠ ∅)
97, 8sylib 221 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) → (𝐹 “ {𝑦}) ≠ ∅)
109adantr 484 . . . . . . . . 9 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) ∧ 𝑧 = (𝐹 “ {𝑦})) → (𝐹 “ {𝑦}) ≠ ∅)
114, 10eqnetrd 3081 . . . . . . . 8 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) ∧ 𝑧 = (𝐹 “ {𝑦})) → 𝑧 ≠ ∅)
1211r19.29an 3280 . . . . . . 7 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ ∃𝑦𝐵 𝑧 = (𝐹 “ {𝑦})) → 𝑧 ≠ ∅)
133, 12sylan2b 596 . . . . . 6 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) → 𝑧 ≠ ∅)
1413ralrimiva 3177 . . . . 5 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))𝑧 ≠ ∅)
15 simp2 1134 . . . . . . . . . . 11 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → 𝐹 Fn 𝐴)
16 simp1 1133 . . . . . . . . . . 11 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → 𝐴𝑉)
1715, 16jca 515 . . . . . . . . . 10 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → (𝐹 Fn 𝐴𝐴𝑉))
18 fnex 6971 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
19 rnexg 7609 . . . . . . . . . 10 (𝐹 ∈ V → ran 𝐹 ∈ V)
2017, 18, 193syl 18 . . . . . . . . 9 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ran 𝐹 ∈ V)
21 simp3 1135 . . . . . . . . 9 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → 𝐵 ⊆ ran 𝐹)
2220, 21ssexd 5214 . . . . . . . 8 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → 𝐵 ∈ V)
23 mptexg 6975 . . . . . . . 8 (𝐵 ∈ V → (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∈ V)
24 rnexg 7609 . . . . . . . 8 ((𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∈ V → ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∈ V)
2522, 23, 243syl 18 . . . . . . 7 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∈ V)
26 fvi 6731 . . . . . . 7 (ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∈ V → ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) = ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
2725, 26syl 17 . . . . . 6 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) = ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
2827raleqdv 3402 . . . . 5 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → (∀𝑧 ∈ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))𝑧 ≠ ∅ ↔ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))𝑧 ≠ ∅))
2914, 28mpbird 260 . . . 4 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ∀𝑧 ∈ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))𝑧 ≠ ∅)
30 fvex 6674 . . . . 5 ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∈ V
3130ac5b 9898 . . . 4 (∀𝑧 ∈ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))𝑧 ≠ ∅ → ∃𝑓(𝑓:( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))⟶ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))(𝑓𝑧) ∈ 𝑧))
3229, 31syl 17 . . 3 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ∃𝑓(𝑓:( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))⟶ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))(𝑓𝑧) ∈ 𝑧))
3327unieqd 4838 . . . . . 6 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) = ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
3427, 33feq23d 6498 . . . . 5 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → (𝑓:( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))⟶ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ↔ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))))
3527raleqdv 3402 . . . . 5 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → (∀𝑧 ∈ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))(𝑓𝑧) ∈ 𝑧 ↔ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧))
3634, 35anbi12d 633 . . . 4 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ((𝑓:( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))⟶ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))(𝑓𝑧) ∈ 𝑧) ↔ (𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧)))
3736exbidv 1923 . . 3 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → (∃𝑓(𝑓:( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))⟶ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ( I ‘ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))(𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓(𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧)))
3832, 37mpbid 235 . 2 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ∃𝑓(𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧))
39 vex 3483 . . . . . . . . 9 𝑓 ∈ V
4039rnex 7612 . . . . . . . 8 ran 𝑓 ∈ V
4140a1i 11 . . . . . . 7 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ran 𝑓 ∈ V)
42 simplr 768 . . . . . . . . 9 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
43 frn 6509 . . . . . . . . 9 (𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) → ran 𝑓 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
4442, 43syl 17 . . . . . . . 8 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ran 𝑓 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
45 nfv 1916 . . . . . . . . . . . . 13 𝑦(𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹)
46 nfcv 2982 . . . . . . . . . . . . . 14 𝑦𝑓
47 nfmpt1 5150 . . . . . . . . . . . . . . 15 𝑦(𝑦𝐵 ↦ (𝐹 “ {𝑦}))
4847nfrn 5811 . . . . . . . . . . . . . 14 𝑦ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))
4948nfuni 4831 . . . . . . . . . . . . . 14 𝑦 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))
5046, 48, 49nff 6499 . . . . . . . . . . . . 13 𝑦 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))
5145, 50nfan 1901 . . . . . . . . . . . 12 𝑦((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
52 nfv 1916 . . . . . . . . . . . . 13 𝑦(𝑓𝑧) ∈ 𝑧
5348, 52nfralw 3219 . . . . . . . . . . . 12 𝑦𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧
5451, 53nfan 1901 . . . . . . . . . . 11 𝑦(((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧)
5517, 18syl 17 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → 𝐹 ∈ V)
5655ad3antrrr 729 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) → 𝐹 ∈ V)
57 cnvexg 7624 . . . . . . . . . . . . . 14 (𝐹 ∈ V → 𝐹 ∈ V)
58 imaexg 7615 . . . . . . . . . . . . . 14 (𝐹 ∈ V → (𝐹 “ {𝑦}) ∈ V)
5956, 57, 583syl 18 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) → (𝐹 “ {𝑦}) ∈ V)
60 cnvimass 5936 . . . . . . . . . . . . . . 15 (𝐹 “ {𝑦}) ⊆ dom 𝐹
6160a1i 11 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) → (𝐹 “ {𝑦}) ⊆ dom 𝐹)
6215fndmd 6445 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → dom 𝐹 = 𝐴)
6362ad3antrrr 729 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) → dom 𝐹 = 𝐴)
6461, 63sseqtrd 3993 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) → (𝐹 “ {𝑦}) ⊆ 𝐴)
6559, 64elpwd 4530 . . . . . . . . . . . 12 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) → (𝐹 “ {𝑦}) ∈ 𝒫 𝐴)
6665ex 416 . . . . . . . . . . 11 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝑦𝐵 → (𝐹 “ {𝑦}) ∈ 𝒫 𝐴))
6754, 66ralrimi 3210 . . . . . . . . . 10 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ∀𝑦𝐵 (𝐹 “ {𝑦}) ∈ 𝒫 𝐴)
681rnmptss 6877 . . . . . . . . . 10 (∀𝑦𝐵 (𝐹 “ {𝑦}) ∈ 𝒫 𝐴 → ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ⊆ 𝒫 𝐴)
6967, 68syl 17 . . . . . . . . 9 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ⊆ 𝒫 𝐴)
70 sspwuni 5008 . . . . . . . . 9 (ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ⊆ 𝒫 𝐴 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ⊆ 𝐴)
7169, 70sylib 221 . . . . . . . 8 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ⊆ 𝐴)
7244, 71sstrd 3963 . . . . . . 7 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ran 𝑓𝐴)
7341, 72elpwd 4530 . . . . . 6 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ran 𝑓 ∈ 𝒫 𝐴)
74 fnfun 6441 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn 𝐴 → Fun 𝐹)
7515, 74syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → Fun 𝐹)
7675ad5antr 733 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ (𝑓𝑢) = (𝑓𝑣)) → Fun 𝐹)
77 sndisj 5043 . . . . . . . . . . . . . . . . . . 19 Disj 𝑦𝐵 {𝑦}
78 disjpreima 30345 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹Disj 𝑦𝐵 {𝑦}) → Disj 𝑦𝐵 (𝐹 “ {𝑦}))
7976, 77, 78sylancl 589 . . . . . . . . . . . . . . . . . 18 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ (𝑓𝑢) = (𝑓𝑣)) → Disj 𝑦𝐵 (𝐹 “ {𝑦}))
80 disjrnmpt 30346 . . . . . . . . . . . . . . . . . 18 (Disj 𝑦𝐵 (𝐹 “ {𝑦}) → Disj 𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))𝑧)
8179, 80syl 17 . . . . . . . . . . . . . . . . 17 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ (𝑓𝑢) = (𝑓𝑣)) → Disj 𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))𝑧)
82 simpllr 775 . . . . . . . . . . . . . . . . 17 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ (𝑓𝑢) = (𝑓𝑣)) → 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
83 simplr 768 . . . . . . . . . . . . . . . . 17 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ (𝑓𝑢) = (𝑓𝑣)) → 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
84 simp-4r 783 . . . . . . . . . . . . . . . . . 18 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ (𝑓𝑢) = (𝑓𝑣)) → ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧)
85 fveq2 6661 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑢 → (𝑓𝑧) = (𝑓𝑢))
86 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑢𝑧 = 𝑢)
8785, 86eleq12d 2910 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝑢 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓𝑢) ∈ 𝑢))
8887rspcv 3604 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) → (∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧 → (𝑓𝑢) ∈ 𝑢))
8988imp 410 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝑓𝑢) ∈ 𝑢)
9082, 84, 89syl2anc 587 . . . . . . . . . . . . . . . . 17 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ (𝑓𝑢) = (𝑓𝑣)) → (𝑓𝑢) ∈ 𝑢)
91 simpr 488 . . . . . . . . . . . . . . . . . 18 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ (𝑓𝑢) = (𝑓𝑣)) → (𝑓𝑢) = (𝑓𝑣))
92 fveq2 6661 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝑣 → (𝑓𝑧) = (𝑓𝑣))
93 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝑣𝑧 = 𝑣)
9492, 93eleq12d 2910 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑣 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓𝑣) ∈ 𝑣))
9594rspcv 3604 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) → (∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧 → (𝑓𝑣) ∈ 𝑣))
9695imp 410 . . . . . . . . . . . . . . . . . . 19 ((𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝑓𝑣) ∈ 𝑣)
9783, 84, 96syl2anc 587 . . . . . . . . . . . . . . . . . 18 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ (𝑓𝑢) = (𝑓𝑣)) → (𝑓𝑣) ∈ 𝑣)
9891, 97eqeltrd 2916 . . . . . . . . . . . . . . . . 17 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ (𝑓𝑢) = (𝑓𝑣)) → (𝑓𝑢) ∈ 𝑣)
9986, 93disji 5035 . . . . . . . . . . . . . . . . 17 ((Disj 𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))𝑧 ∧ (𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ((𝑓𝑢) ∈ 𝑢 ∧ (𝑓𝑢) ∈ 𝑣)) → 𝑢 = 𝑣)
10081, 82, 83, 90, 98, 99syl122anc 1376 . . . . . . . . . . . . . . . 16 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ (𝑓𝑢) = (𝑓𝑣)) → 𝑢 = 𝑣)
101100ex 416 . . . . . . . . . . . . . . 15 ((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) → ((𝑓𝑢) = (𝑓𝑣) → 𝑢 = 𝑣))
102101anasss 470 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ (𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ 𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))) → ((𝑓𝑢) = (𝑓𝑣) → 𝑢 = 𝑣))
103102ralrimivva 3186 . . . . . . . . . . . . 13 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ∀𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))∀𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))((𝑓𝑢) = (𝑓𝑣) → 𝑢 = 𝑣))
10442, 103jca 515 . . . . . . . . . . . 12 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ ∀𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))∀𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))((𝑓𝑢) = (𝑓𝑣) → 𝑢 = 𝑣)))
105 dff13 7005 . . . . . . . . . . . 12 (𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))–1-1 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ↔ (𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ ∀𝑢 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))∀𝑣 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))((𝑓𝑢) = (𝑓𝑣) → 𝑢 = 𝑣)))
106104, 105sylibr 237 . . . . . . . . . . 11 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))–1-1 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
107 f1f1orn 6617 . . . . . . . . . . 11 (𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))–1-1 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) → 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))–1-1-onto→ran 𝑓)
108106, 107syl 17 . . . . . . . . . 10 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))–1-1-onto→ran 𝑓)
109 f1oen3g 8521 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))–1-1-onto→ran 𝑓) → ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ≈ ran 𝑓)
11039, 108, 109sylancr 590 . . . . . . . . 9 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ≈ ran 𝑓)
111110ensymd 8556 . . . . . . . 8 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ran 𝑓 ≈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
11222, 23syl 17 . . . . . . . . . . 11 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∈ V)
113112ad2antrr 725 . . . . . . . . . 10 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∈ V)
11459ex 416 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝑦𝐵 → (𝐹 “ {𝑦}) ∈ V))
11554, 114ralrimi 3210 . . . . . . . . . . . . 13 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ∀𝑦𝐵 (𝐹 “ {𝑦}) ∈ V)
11675ad5antr 733 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) ∧ 𝑡𝐵) ∧ 𝑦𝑡) → Fun 𝐹)
117 simpr 488 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) ∧ 𝑡𝐵) ∧ 𝑦𝑡) → 𝑦𝑡)
11821ad5antr 733 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) ∧ 𝑡𝐵) ∧ 𝑦𝑡) → 𝐵 ⊆ ran 𝐹)
119 simpllr 775 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) ∧ 𝑡𝐵) ∧ 𝑦𝑡) → 𝑦𝐵)
120118, 119sseldd 3954 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) ∧ 𝑡𝐵) ∧ 𝑦𝑡) → 𝑦 ∈ ran 𝐹)
121 simplr 768 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) ∧ 𝑡𝐵) ∧ 𝑦𝑡) → 𝑡𝐵)
122118, 121sseldd 3954 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) ∧ 𝑡𝐵) ∧ 𝑦𝑡) → 𝑡 ∈ ran 𝐹)
123116, 117, 120, 122preimane 30426 . . . . . . . . . . . . . . . . . 18 (((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) ∧ 𝑡𝐵) ∧ 𝑦𝑡) → (𝐹 “ {𝑦}) ≠ (𝐹 “ {𝑡}))
124123ex 416 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) ∧ 𝑡𝐵) → (𝑦𝑡 → (𝐹 “ {𝑦}) ≠ (𝐹 “ {𝑡})))
125124necon4d 3038 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) ∧ 𝑡𝐵) → ((𝐹 “ {𝑦}) = (𝐹 “ {𝑡}) → 𝑦 = 𝑡))
126125ralrimiva 3177 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑦𝐵) → ∀𝑡𝐵 ((𝐹 “ {𝑦}) = (𝐹 “ {𝑡}) → 𝑦 = 𝑡))
127126ex 416 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝑦𝐵 → ∀𝑡𝐵 ((𝐹 “ {𝑦}) = (𝐹 “ {𝑡}) → 𝑦 = 𝑡)))
12854, 127ralrimi 3210 . . . . . . . . . . . . 13 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ∀𝑦𝐵𝑡𝐵 ((𝐹 “ {𝑦}) = (𝐹 “ {𝑡}) → 𝑦 = 𝑡))
129115, 128jca 515 . . . . . . . . . . . 12 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (∀𝑦𝐵 (𝐹 “ {𝑦}) ∈ V ∧ ∀𝑦𝐵𝑡𝐵 ((𝐹 “ {𝑦}) = (𝐹 “ {𝑡}) → 𝑦 = 𝑡)))
130 sneq 4560 . . . . . . . . . . . . . 14 (𝑦 = 𝑡 → {𝑦} = {𝑡})
131130imaeq2d 5916 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → (𝐹 “ {𝑦}) = (𝐹 “ {𝑡}))
1321, 131f1mpt 7011 . . . . . . . . . . . 12 ((𝑦𝐵 ↦ (𝐹 “ {𝑦})):𝐵1-1→V ↔ (∀𝑦𝐵 (𝐹 “ {𝑦}) ∈ V ∧ ∀𝑦𝐵𝑡𝐵 ((𝐹 “ {𝑦}) = (𝐹 “ {𝑡}) → 𝑦 = 𝑡)))
133129, 132sylibr 237 . . . . . . . . . . 11 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝑦𝐵 ↦ (𝐹 “ {𝑦})):𝐵1-1→V)
134 f1f1orn 6617 . . . . . . . . . . 11 ((𝑦𝐵 ↦ (𝐹 “ {𝑦})):𝐵1-1→V → (𝑦𝐵 ↦ (𝐹 “ {𝑦})):𝐵1-1-onto→ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
135133, 134syl 17 . . . . . . . . . 10 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝑦𝐵 ↦ (𝐹 “ {𝑦})):𝐵1-1-onto→ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
136 f1oen3g 8521 . . . . . . . . . 10 (((𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∈ V ∧ (𝑦𝐵 ↦ (𝐹 “ {𝑦})):𝐵1-1-onto→ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) → 𝐵 ≈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
137113, 135, 136syl2anc 587 . . . . . . . . 9 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → 𝐵 ≈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
138137ensymd 8556 . . . . . . . 8 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ≈ 𝐵)
139 entr 8557 . . . . . . . 8 ((ran 𝑓 ≈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ≈ 𝐵) → ran 𝑓𝐵)
140111, 138, 139syl2anc 587 . . . . . . 7 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ran 𝑓𝐵)
141 imass2 5952 . . . . . . . . . . 11 (ran 𝑓 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) → (𝐹 “ ran 𝑓) ⊆ (𝐹 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))))
14243, 141syl 17 . . . . . . . . . 10 (𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) → (𝐹 “ ran 𝑓) ⊆ (𝐹 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))))
14342, 142syl 17 . . . . . . . . 9 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝐹 “ ran 𝑓) ⊆ (𝐹 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))))
144 imauni 6997 . . . . . . . . . 10 (𝐹 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) = 𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝐹𝑧)
145 imaeq2 5912 . . . . . . . . . . . . 13 (𝑧 = (𝐹 “ {𝑦}) → (𝐹𝑧) = (𝐹 “ (𝐹 “ {𝑦})))
14655adantr 484 . . . . . . . . . . . . . 14 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) → 𝐹 ∈ V)
147146, 57, 583syl 18 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) → (𝐹 “ {𝑦}) ∈ V)
148145, 147iunrnmptss 30328 . . . . . . . . . . . 12 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → 𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝐹𝑧) ⊆ 𝑦𝐵 (𝐹 “ (𝐹 “ {𝑦})))
149 funimacnv 6423 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 → (𝐹 “ (𝐹 “ {𝑦})) = ({𝑦} ∩ ran 𝐹))
15075, 149syl 17 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → (𝐹 “ (𝐹 “ {𝑦})) = ({𝑦} ∩ ran 𝐹))
151150adantr 484 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) → (𝐹 “ (𝐹 “ {𝑦})) = ({𝑦} ∩ ran 𝐹))
1526snssd 4726 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) → {𝑦} ⊆ 𝐵)
153152, 5sstrd 3963 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) → {𝑦} ⊆ ran 𝐹)
154 df-ss 3936 . . . . . . . . . . . . . . . 16 ({𝑦} ⊆ ran 𝐹 ↔ ({𝑦} ∩ ran 𝐹) = {𝑦})
155153, 154sylib 221 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) → ({𝑦} ∩ ran 𝐹) = {𝑦})
156151, 155eqtrd 2859 . . . . . . . . . . . . . 14 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑦𝐵) → (𝐹 “ (𝐹 “ {𝑦})) = {𝑦})
157156iuneq2dv 4929 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → 𝑦𝐵 (𝐹 “ (𝐹 “ {𝑦})) = 𝑦𝐵 {𝑦})
158 iunid 4970 . . . . . . . . . . . . 13 𝑦𝐵 {𝑦} = 𝐵
159157, 158syl6eq 2875 . . . . . . . . . . . 12 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → 𝑦𝐵 (𝐹 “ (𝐹 “ {𝑦})) = 𝐵)
160148, 159sseqtrd 3993 . . . . . . . . . . 11 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → 𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝐹𝑧) ⊆ 𝐵)
161160ad2antrr 725 . . . . . . . . . 10 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → 𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝐹𝑧) ⊆ 𝐵)
162144, 161eqsstrid 4001 . . . . . . . . 9 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝐹 ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ⊆ 𝐵)
163143, 162sstrd 3963 . . . . . . . 8 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝐹 “ ran 𝑓) ⊆ 𝐵)
16442adantr 484 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
165164ffund 6507 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → Fun 𝑓)
166 simpr 488 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → 𝑡𝐵)
16755, 57syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → 𝐹 ∈ V)
168167ad3antrrr 729 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → 𝐹 ∈ V)
169 imaexg 7615 . . . . . . . . . . . . . . . 16 (𝐹 ∈ V → (𝐹 “ {𝑡}) ∈ V)
170168, 169syl 17 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → (𝐹 “ {𝑡}) ∈ V)
1711, 131elrnmpt1s 5816 . . . . . . . . . . . . . . 15 ((𝑡𝐵 ∧ (𝐹 “ {𝑡}) ∈ V) → (𝐹 “ {𝑡}) ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
172166, 170, 171syl2anc 587 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → (𝐹 “ {𝑡}) ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
173164fdmd 6513 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → dom 𝑓 = ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})))
174172, 173eleqtrrd 2919 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → (𝐹 “ {𝑡}) ∈ dom 𝑓)
175 fvelrn 6835 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ (𝐹 “ {𝑡}) ∈ dom 𝑓) → (𝑓‘(𝐹 “ {𝑡})) ∈ ran 𝑓)
176165, 174, 175syl2anc 587 . . . . . . . . . . . 12 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → (𝑓‘(𝐹 “ {𝑡})) ∈ ran 𝑓)
17715ad3antrrr 729 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → 𝐹 Fn 𝐴)
178 simplr 768 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧)
179 fveq2 6661 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐹 “ {𝑡}) → (𝑓𝑧) = (𝑓‘(𝐹 “ {𝑡})))
180 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐹 “ {𝑡}) → 𝑧 = (𝐹 “ {𝑡}))
181179, 180eleq12d 2910 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹 “ {𝑡}) → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓‘(𝐹 “ {𝑡})) ∈ (𝐹 “ {𝑡})))
182181rspcv 3604 . . . . . . . . . . . . . . 15 ((𝐹 “ {𝑡}) ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) → (∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧 → (𝑓‘(𝐹 “ {𝑡})) ∈ (𝐹 “ {𝑡})))
183182imp 410 . . . . . . . . . . . . . 14 (((𝐹 “ {𝑡}) ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝑓‘(𝐹 “ {𝑡})) ∈ (𝐹 “ {𝑡}))
184172, 178, 183syl2anc 587 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → (𝑓‘(𝐹 “ {𝑡})) ∈ (𝐹 “ {𝑡}))
185 fniniseg 6821 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → ((𝑓‘(𝐹 “ {𝑡})) ∈ (𝐹 “ {𝑡}) ↔ ((𝑓‘(𝐹 “ {𝑡})) ∈ 𝐴 ∧ (𝐹‘(𝑓‘(𝐹 “ {𝑡}))) = 𝑡)))
186185simplbda 503 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴 ∧ (𝑓‘(𝐹 “ {𝑡})) ∈ (𝐹 “ {𝑡})) → (𝐹‘(𝑓‘(𝐹 “ {𝑡}))) = 𝑡)
187177, 184, 186syl2anc 587 . . . . . . . . . . . 12 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → (𝐹‘(𝑓‘(𝐹 “ {𝑡}))) = 𝑡)
188 fveqeq2 6670 . . . . . . . . . . . . 13 (𝑘 = (𝑓‘(𝐹 “ {𝑡})) → ((𝐹𝑘) = 𝑡 ↔ (𝐹‘(𝑓‘(𝐹 “ {𝑡}))) = 𝑡))
189188rspcev 3609 . . . . . . . . . . . 12 (((𝑓‘(𝐹 “ {𝑡})) ∈ ran 𝑓 ∧ (𝐹‘(𝑓‘(𝐹 “ {𝑡}))) = 𝑡) → ∃𝑘 ∈ ran 𝑓(𝐹𝑘) = 𝑡)
190176, 187, 189syl2anc 587 . . . . . . . . . . 11 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → ∃𝑘 ∈ ran 𝑓(𝐹𝑘) = 𝑡)
19172adantr 484 . . . . . . . . . . . 12 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → ran 𝑓𝐴)
192177, 191fvelimabd 6729 . . . . . . . . . . 11 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → (𝑡 ∈ (𝐹 “ ran 𝑓) ↔ ∃𝑘 ∈ ran 𝑓(𝐹𝑘) = 𝑡))
193190, 192mpbird 260 . . . . . . . . . 10 (((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) ∧ 𝑡𝐵) → 𝑡 ∈ (𝐹 “ ran 𝑓))
194193ex 416 . . . . . . . . 9 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝑡𝐵𝑡 ∈ (𝐹 “ ran 𝑓)))
195194ssrdv 3959 . . . . . . . 8 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → 𝐵 ⊆ (𝐹 “ ran 𝑓))
196163, 195eqssd 3970 . . . . . . 7 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (𝐹 “ ran 𝑓) = 𝐵)
197140, 196jca 515 . . . . . 6 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → (ran 𝑓𝐵 ∧ (𝐹 “ ran 𝑓) = 𝐵))
198 breq1 5055 . . . . . . . 8 (𝑥 = ran 𝑓 → (𝑥𝐵 ↔ ran 𝑓𝐵))
199 imaeq2 5912 . . . . . . . . 9 (𝑥 = ran 𝑓 → (𝐹𝑥) = (𝐹 “ ran 𝑓))
200199eqeq1d 2826 . . . . . . . 8 (𝑥 = ran 𝑓 → ((𝐹𝑥) = 𝐵 ↔ (𝐹 “ ran 𝑓) = 𝐵))
201198, 200anbi12d 633 . . . . . . 7 (𝑥 = ran 𝑓 → ((𝑥𝐵 ∧ (𝐹𝑥) = 𝐵) ↔ (ran 𝑓𝐵 ∧ (𝐹 “ ran 𝑓) = 𝐵)))
202201rspcev 3609 . . . . . 6 ((ran 𝑓 ∈ 𝒫 𝐴 ∧ (ran 𝑓𝐵 ∧ (𝐹 “ ran 𝑓) = 𝐵)) → ∃𝑥 ∈ 𝒫 𝐴(𝑥𝐵 ∧ (𝐹𝑥) = 𝐵))
20373, 197, 202syl2anc 587 . . . . 5 ((((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ 𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ∃𝑥 ∈ 𝒫 𝐴(𝑥𝐵 ∧ (𝐹𝑥) = 𝐵))
204203anasss 470 . . . 4 (((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) ∧ (𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧)) → ∃𝑥 ∈ 𝒫 𝐴(𝑥𝐵 ∧ (𝐹𝑥) = 𝐵))
205204ex 416 . . 3 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ((𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ∃𝑥 ∈ 𝒫 𝐴(𝑥𝐵 ∧ (𝐹𝑥) = 𝐵)))
206205exlimdv 1935 . 2 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → (∃𝑓(𝑓:ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))⟶ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦})) ∧ ∀𝑧 ∈ ran (𝑦𝐵 ↦ (𝐹 “ {𝑦}))(𝑓𝑧) ∈ 𝑧) → ∃𝑥 ∈ 𝒫 𝐴(𝑥𝐵 ∧ (𝐹𝑥) = 𝐵)))
20738, 206mpd 15 1 ((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ∃𝑥 ∈ 𝒫 𝐴(𝑥𝐵 ∧ (𝐹𝑥) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  ∃wrex 3134  Vcvv 3480   ∩ cin 3918   ⊆ wss 3919  ∅c0 4276  𝒫 cpw 4522  {csn 4550  ∪ cuni 4824  ∪ ciun 4905  Disj wdisj 5017   class class class wbr 5052   ↦ cmpt 5132   I cid 5446  ◡ccnv 5541  dom cdm 5542  ran crn 5543   “ cima 5545  Fun wfun 6337   Fn wfn 6338  ⟶wf 6339  –1-1→wf1 6340  –1-1-onto→wf1o 6342  ‘cfv 6343   ≈ cen 8502 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-ac2 9883 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-disj 5018  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-wrecs 7943  df-recs 8004  df-er 8285  df-en 8506  df-card 9365  df-ac 9540 This theorem is referenced by: (None)
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