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Theorem tz7.48lem 7496
 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 2935 . . . . . . 7 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
2 simpl 473 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐴) → 𝑥𝐴)
32anim1i 591 . . . . . . . . . 10 (((𝑥𝐴𝑦𝐴) ∧ 𝑦𝑥) → (𝑥𝐴𝑦𝑥))
43imim1i 63 . . . . . . . . 9 (((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)) → (((𝑥𝐴𝑦𝐴) ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
54expd 452 . . . . . . . 8 (((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)) → ((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
652alimi 1737 . . . . . . 7 (∀𝑥𝑦((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)) → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
71, 6sylbi 207 . . . . . 6 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
8 r2al 2935 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
97, 8sylibr 224 . . . . 5 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
10 elequ1 1994 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑦𝑥𝑤𝑥))
11 fveq2 6158 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
1211eqeq2d 2631 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑥) = (𝐹𝑤)))
1312notbid 308 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (¬ (𝐹𝑥) = (𝐹𝑦) ↔ ¬ (𝐹𝑥) = (𝐹𝑤)))
1410, 13imbi12d 334 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤))))
1514cbvralv 3163 . . . . . . . . . 10 (∀𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)))
1615ralbii 2976 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑥𝐴𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)))
17 elequ2 2001 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑤𝑥𝑤𝑧))
18 fveq2 6158 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
1918eqeq1d 2623 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑥) = (𝐹𝑤) ↔ (𝐹𝑧) = (𝐹𝑤)))
2019notbid 308 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (¬ (𝐹𝑥) = (𝐹𝑤) ↔ ¬ (𝐹𝑧) = (𝐹𝑤)))
2117, 20imbi12d 334 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)) ↔ (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤))))
2221ralbidv 2982 . . . . . . . . . 10 (𝑥 = 𝑧 → (∀𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)) ↔ ∀𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤))))
2322cbvralv 3163 . . . . . . . . 9 (∀𝑥𝐴𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)) ↔ ∀𝑧𝐴𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)))
24 elequ1 1994 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
25 fveq2 6158 . . . . . . . . . . . . . . 15 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
2625eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑧) = (𝐹𝑥)))
2726notbid 308 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (¬ (𝐹𝑧) = (𝐹𝑤) ↔ ¬ (𝐹𝑧) = (𝐹𝑥)))
2824, 27imbi12d 334 . . . . . . . . . . . 12 (𝑤 = 𝑥 → ((𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥))))
2928cbvralv 3163 . . . . . . . . . . 11 (∀𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ ∀𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)))
3029ralbii 2976 . . . . . . . . . 10 (∀𝑧𝐴𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ ∀𝑧𝐴𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)))
31 elequ2 2001 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
32 fveq2 6158 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
3332eqeq1d 2623 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → ((𝐹𝑧) = (𝐹𝑥) ↔ (𝐹𝑦) = (𝐹𝑥)))
3433notbid 308 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (¬ (𝐹𝑧) = (𝐹𝑥) ↔ ¬ (𝐹𝑦) = (𝐹𝑥)))
3531, 34imbi12d 334 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ((𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)) ↔ (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥))))
3635ralbidv 2982 . . . . . . . . . . 11 (𝑧 = 𝑦 → (∀𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)) ↔ ∀𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥))))
3736cbvralv 3163 . . . . . . . . . 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 3098 . . . . . . . 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 3060 . . . . . 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 6174 . . . . . . . . . . 11 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
47 fvres 6174 . . . . . . . . . . 11 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
4846, 47eqeqan12d 2637 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
4948ad2antrl 763 . . . . . . . . 9 ((𝐴 ⊆ On ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
50 ssel 3582 . . . . . . . . . . . 12 (𝐴 ⊆ On → (𝑥𝐴𝑥 ∈ On))
51 ssel 3582 . . . . . . . . . . . 12 (𝐴 ⊆ On → (𝑦𝐴𝑦 ∈ On))
5250, 51anim12d 585 . . . . . . . . . . 11 (𝐴 ⊆ On → ((𝑥𝐴𝑦𝐴) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
53 pm3.48 877 . . . . . . . . . . . . . 14 (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝑥𝑦𝑦𝑥) → (¬ (𝐹𝑦) = (𝐹𝑥) ∨ ¬ (𝐹𝑥) = (𝐹𝑦))))
54 oridm 536 . . . . . . . . . . . . . . 15 ((¬ (𝐹𝑥) = (𝐹𝑦) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ¬ (𝐹𝑥) = (𝐹𝑦))
55 eqcom 2628 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
5655notbii 310 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑥) = (𝐹𝑦) ↔ ¬ (𝐹𝑦) = (𝐹𝑥))
5756orbi1i 542 . . . . . . . . . . . . . . 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 5702 . . . . . . . . . . . . 13 (𝑥 ∈ On → Ord 𝑥)
62 eloni 5702 . . . . . . . . . . . . 13 (𝑦 ∈ On → Ord 𝑦)
63 ordtri3 5728 . . . . . . . . . . . . . 14 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 = 𝑦 ↔ ¬ (𝑥𝑦𝑦𝑥)))
6463biimprd 238 . . . . . . . . . . . . 13 ((Ord 𝑥 ∧ Ord 𝑦) → (¬ (𝑥𝑦𝑦𝑥) → 𝑥 = 𝑦))
6561, 62, 64syl2an 494 . . . . . . . . . . . 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 449 . . . . . . . . 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 1844 . . . . 5 (𝐴 ⊆ On → (∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))) → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))))
73 r2al 2935 . . . . 5 (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))))
74 r2al 2935 . . . . 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 725 . 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 5972 . . . 4 ((𝐹 Fn On ∧ 𝐴 ⊆ On) → (𝐹𝐴) Fn 𝐴)
8078, 79mpan 705 . . 3 (𝐴 ⊆ On → (𝐹𝐴) Fn 𝐴)
81 dffn2 6014 . . . 4 ((𝐹𝐴) Fn 𝐴 ↔ (𝐹𝐴):𝐴⟶V)
82 dff13 6477 . . . . . 6 ((𝐹𝐴):𝐴1-1→V ↔ ((𝐹𝐴):𝐴⟶V ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
83 df-f1 5862 . . . . . 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 480 . . . 4 (((𝐹𝐴):𝐴⟶V ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun (𝐹𝐴))
8681, 85sylanb 489 . . 3 (((𝐹𝐴) Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun (𝐹𝐴))
8780, 86sylan 488 . 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 383   ∧ wa 384  ∀wal 1478   = wceq 1480   ∈ wcel 1987  ∀wral 2908  Vcvv 3190   ⊆ wss 3560  ◡ccnv 5083   ↾ cres 5086  Ord word 5691  Oncon0 5692  Fun wfun 5851   Fn wfn 5852  ⟶wf 5853  –1-1→wf1 5854  ‘cfv 5857 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-res 5096  df-ord 5695  df-on 5696  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fv 5865 This theorem is referenced by:  tz7.48-2  7497  tz7.49  7500  zorn2lem4  9281
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