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Theorem tz7.48lem 7581
Description: A way of showing an ordinal function is one-to-one. (Contributed by NM, 9-Feb-1997.)
Hypothesis
Ref Expression
tz7.48.1 𝐹 Fn On
Assertion
Ref Expression
tz7.48lem ((𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹𝐴))
Distinct variable groups:   𝑦,𝐴,𝑥   𝑥,𝐹,𝑦   𝑥,𝐴

Proof of Theorem tz7.48lem
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r2al 2968 . . . . . . 7 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
2 simpl 472 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐴) → 𝑥𝐴)
32anim1i 591 . . . . . . . . . 10 (((𝑥𝐴𝑦𝐴) ∧ 𝑦𝑥) → (𝑥𝐴𝑦𝑥))
43imim1i 63 . . . . . . . . 9 (((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)) → (((𝑥𝐴𝑦𝐴) ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
54expd 451 . . . . . . . 8 (((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)) → ((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
652alimi 1780 . . . . . . 7 (∀𝑥𝑦((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)) → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
71, 6sylbi 207 . . . . . 6 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
8 r2al 2968 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
97, 8sylibr 224 . . . . 5 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
10 elequ1 2037 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑦𝑥𝑤𝑥))
11 fveq2 6229 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
1211eqeq2d 2661 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑥) = (𝐹𝑤)))
1312notbid 307 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (¬ (𝐹𝑥) = (𝐹𝑦) ↔ ¬ (𝐹𝑥) = (𝐹𝑤)))
1410, 13imbi12d 333 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤))))
1514cbvralv 3201 . . . . . . . . . 10 (∀𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)))
1615ralbii 3009 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑥𝐴𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)))
17 elequ2 2044 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑤𝑥𝑤𝑧))
18 fveq2 6229 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
1918eqeq1d 2653 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑥) = (𝐹𝑤) ↔ (𝐹𝑧) = (𝐹𝑤)))
2019notbid 307 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (¬ (𝐹𝑥) = (𝐹𝑤) ↔ ¬ (𝐹𝑧) = (𝐹𝑤)))
2117, 20imbi12d 333 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)) ↔ (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤))))
2221ralbidv 3015 . . . . . . . . . 10 (𝑥 = 𝑧 → (∀𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)) ↔ ∀𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤))))
2322cbvralv 3201 . . . . . . . . 9 (∀𝑥𝐴𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)) ↔ ∀𝑧𝐴𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)))
24 elequ1 2037 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
25 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
2625eqeq2d 2661 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑧) = (𝐹𝑥)))
2726notbid 307 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (¬ (𝐹𝑧) = (𝐹𝑤) ↔ ¬ (𝐹𝑧) = (𝐹𝑥)))
2824, 27imbi12d 333 . . . . . . . . . . . 12 (𝑤 = 𝑥 → ((𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥))))
2928cbvralv 3201 . . . . . . . . . . 11 (∀𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ ∀𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)))
3029ralbii 3009 . . . . . . . . . 10 (∀𝑧𝐴𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ ∀𝑧𝐴𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)))
31 elequ2 2044 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
32 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
3332eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → ((𝐹𝑧) = (𝐹𝑥) ↔ (𝐹𝑦) = (𝐹𝑥)))
3433notbid 307 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (¬ (𝐹𝑧) = (𝐹𝑥) ↔ ¬ (𝐹𝑦) = (𝐹𝑥)))
3531, 34imbi12d 333 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ((𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)) ↔ (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥))))
3635ralbidv 3015 . . . . . . . . . . 11 (𝑧 = 𝑦 → (∀𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)) ↔ ∀𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥))))
3736cbvralv 3201 . . . . . . . . . 10 (∀𝑧𝐴𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
3830, 37bitri 264 . . . . . . . . 9 (∀𝑧𝐴𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
3916, 23, 383bitri 286 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
40 ralcom2 3133 . . . . . . . 8 (∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
4139, 40sylbi 207 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
4241ancri 574 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ ∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
43 r19.26-2 3094 . . . . . 6 (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) ↔ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ ∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
4442, 43sylibr 224 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
459, 44syl 17 . . . 4 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
46 fvres 6245 . . . . . . . . . . 11 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
47 fvres 6245 . . . . . . . . . . 11 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
4846, 47eqeqan12d 2667 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
4948ad2antrl 764 . . . . . . . . 9 ((𝐴 ⊆ On ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
50 ssel 3630 . . . . . . . . . . . 12 (𝐴 ⊆ On → (𝑥𝐴𝑥 ∈ On))
51 ssel 3630 . . . . . . . . . . . 12 (𝐴 ⊆ On → (𝑦𝐴𝑦 ∈ On))
5250, 51anim12d 585 . . . . . . . . . . 11 (𝐴 ⊆ On → ((𝑥𝐴𝑦𝐴) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
53 pm3.48 896 . . . . . . . . . . . . . 14 (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝑥𝑦𝑦𝑥) → (¬ (𝐹𝑦) = (𝐹𝑥) ∨ ¬ (𝐹𝑥) = (𝐹𝑦))))
54 oridm 535 . . . . . . . . . . . . . . 15 ((¬ (𝐹𝑥) = (𝐹𝑦) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ¬ (𝐹𝑥) = (𝐹𝑦))
55 eqcom 2658 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
5655notbii 309 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑥) = (𝐹𝑦) ↔ ¬ (𝐹𝑦) = (𝐹𝑥))
5756orbi1i 541 . . . . . . . . . . . . . . 15 ((¬ (𝐹𝑥) = (𝐹𝑦) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ (¬ (𝐹𝑦) = (𝐹𝑥) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)))
5854, 57bitr3i 266 . . . . . . . . . . . . . 14 (¬ (𝐹𝑥) = (𝐹𝑦) ↔ (¬ (𝐹𝑦) = (𝐹𝑥) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)))
5953, 58syl6ibr 242 . . . . . . . . . . . . 13 (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝑥𝑦𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
6059con2d 129 . . . . . . . . . . . 12 (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝐹𝑥) = (𝐹𝑦) → ¬ (𝑥𝑦𝑦𝑥)))
61 eloni 5771 . . . . . . . . . . . . 13 (𝑥 ∈ On → Ord 𝑥)
62 eloni 5771 . . . . . . . . . . . . 13 (𝑦 ∈ On → Ord 𝑦)
63 ordtri3 5797 . . . . . . . . . . . . . 14 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 = 𝑦 ↔ ¬ (𝑥𝑦𝑦𝑥)))
6463biimprd 238 . . . . . . . . . . . . 13 ((Ord 𝑥 ∧ Ord 𝑦) → (¬ (𝑥𝑦𝑦𝑥) → 𝑥 = 𝑦))
6561, 62, 64syl2an 493 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (¬ (𝑥𝑦𝑦𝑥) → 𝑥 = 𝑦))
6660, 65syl9r 78 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6752, 66syl6 35 . . . . . . . . . 10 (𝐴 ⊆ On → ((𝑥𝐴𝑦𝐴) → (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
6867imp32 448 . . . . . . . . 9 ((𝐴 ⊆ On ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6949, 68sylbid 230 . . . . . . . 8 ((𝐴 ⊆ On ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))
7069exp32 630 . . . . . . 7 (𝐴 ⊆ On → ((𝑥𝐴𝑦𝐴) → (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))))
7170a2d 29 . . . . . 6 (𝐴 ⊆ On → (((𝑥𝐴𝑦𝐴) → ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))) → ((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))))
72712alimdv 1887 . . . . 5 (𝐴 ⊆ On → (∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))) → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))))
73 r2al 2968 . . . . 5 (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))))
74 r2al 2968 . . . . 5 (∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
7572, 73, 743imtr4g 285 . . . 4 (𝐴 ⊆ On → (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
7645, 75syl5 34 . . 3 (𝐴 ⊆ On → (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
7776imdistani 726 . 2 ((𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
78 tz7.48.1 . . . 4 𝐹 Fn On
79 fnssres 6042 . . . 4 ((𝐹 Fn On ∧ 𝐴 ⊆ On) → (𝐹𝐴) Fn 𝐴)
8078, 79mpan 706 . . 3 (𝐴 ⊆ On → (𝐹𝐴) Fn 𝐴)
81 dffn2 6085 . . . 4 ((𝐹𝐴) Fn 𝐴 ↔ (𝐹𝐴):𝐴⟶V)
82 dff13 6552 . . . . . 6 ((𝐹𝐴):𝐴1-1→V ↔ ((𝐹𝐴):𝐴⟶V ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
83 df-f1 5931 . . . . . 6 ((𝐹𝐴):𝐴1-1→V ↔ ((𝐹𝐴):𝐴⟶V ∧ Fun (𝐹𝐴)))
8482, 83bitr3i 266 . . . . 5 (((𝐹𝐴):𝐴⟶V ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹𝐴):𝐴⟶V ∧ Fun (𝐹𝐴)))
8584simprbi 479 . . . 4 (((𝐹𝐴):𝐴⟶V ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun (𝐹𝐴))
8681, 85sylanb 488 . . 3 (((𝐹𝐴) Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun (𝐹𝐴))
8780, 86sylan 487 . 2 ((𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun (𝐹𝐴))
8877, 87syl 17 1 ((𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  wal 1521   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607  ccnv 5142  cres 5145  Ord word 5760  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  wf 5922  1-1wf1 5923  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934
This theorem is referenced by:  tz7.48-2  7582  tz7.49  7585  zorn2lem4  9359
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