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Theorem tz7.48lem 8481
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 3195 . . . . . . 7 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
2 simpl 482 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐴) → 𝑥𝐴)
32anim1i 615 . . . . . . . . . 10 (((𝑥𝐴𝑦𝐴) ∧ 𝑦𝑥) → (𝑥𝐴𝑦𝑥))
43imim1i 63 . . . . . . . . 9 (((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)) → (((𝑥𝐴𝑦𝐴) ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
54expd 415 . . . . . . . 8 (((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)) → ((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
652alimi 1812 . . . . . . 7 (∀𝑥𝑦((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)) → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
71, 6sylbi 217 . . . . . 6 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
8 r2al 3195 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
97, 8sylibr 234 . . . . 5 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
10 elequ1 2115 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑦𝑥𝑤𝑥))
11 fveq2 6906 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
1211eqeq2d 2748 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑥) = (𝐹𝑤)))
1312notbid 318 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (¬ (𝐹𝑥) = (𝐹𝑦) ↔ ¬ (𝐹𝑥) = (𝐹𝑤)))
1410, 13imbi12d 344 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤))))
1514cbvralvw 3237 . . . . . . . . . 10 (∀𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)))
1615ralbii 3093 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑥𝐴𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)))
17 elequ2 2123 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑤𝑥𝑤𝑧))
18 fveqeq2 6915 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑥) = (𝐹𝑤) ↔ (𝐹𝑧) = (𝐹𝑤)))
1918notbid 318 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (¬ (𝐹𝑥) = (𝐹𝑤) ↔ ¬ (𝐹𝑧) = (𝐹𝑤)))
2017, 19imbi12d 344 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)) ↔ (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤))))
2120ralbidv 3178 . . . . . . . . . 10 (𝑥 = 𝑧 → (∀𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)) ↔ ∀𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤))))
2221cbvralvw 3237 . . . . . . . . 9 (∀𝑥𝐴𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)) ↔ ∀𝑧𝐴𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)))
23 elequ1 2115 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
24 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
2524eqeq2d 2748 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑧) = (𝐹𝑥)))
2625notbid 318 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (¬ (𝐹𝑧) = (𝐹𝑤) ↔ ¬ (𝐹𝑧) = (𝐹𝑥)))
2723, 26imbi12d 344 . . . . . . . . . . . 12 (𝑤 = 𝑥 → ((𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥))))
2827cbvralvw 3237 . . . . . . . . . . 11 (∀𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ ∀𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)))
2928ralbii 3093 . . . . . . . . . 10 (∀𝑧𝐴𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ ∀𝑧𝐴𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)))
30 elequ2 2123 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
31 fveqeq2 6915 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → ((𝐹𝑧) = (𝐹𝑥) ↔ (𝐹𝑦) = (𝐹𝑥)))
3231notbid 318 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (¬ (𝐹𝑧) = (𝐹𝑥) ↔ ¬ (𝐹𝑦) = (𝐹𝑥)))
3330, 32imbi12d 344 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ((𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)) ↔ (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥))))
3433ralbidv 3178 . . . . . . . . . . 11 (𝑧 = 𝑦 → (∀𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)) ↔ ∀𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥))))
3534cbvralvw 3237 . . . . . . . . . 10 (∀𝑧𝐴𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
3629, 35bitri 275 . . . . . . . . 9 (∀𝑧𝐴𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
3716, 22, 363bitri 297 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
38 ralcom 3289 . . . . . . . . 9 (∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
3938biimpi 216 . . . . . . . 8 (∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
4037, 39sylbi 217 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
4140ancri 549 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ ∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
42 r19.26-2 3138 . . . . . 6 (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) ↔ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ ∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
4341, 42sylibr 234 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
449, 43syl 17 . . . 4 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
45 fvres 6925 . . . . . . . . . . 11 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
46 fvres 6925 . . . . . . . . . . 11 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
4745, 46eqeqan12d 2751 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
4847ad2antrl 728 . . . . . . . . 9 ((𝐴 ⊆ On ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
49 ssel 3977 . . . . . . . . . . . 12 (𝐴 ⊆ On → (𝑥𝐴𝑥 ∈ On))
50 ssel 3977 . . . . . . . . . . . 12 (𝐴 ⊆ On → (𝑦𝐴𝑦 ∈ On))
5149, 50anim12d 609 . . . . . . . . . . 11 (𝐴 ⊆ On → ((𝑥𝐴𝑦𝐴) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
52 pm3.48 966 . . . . . . . . . . . . . 14 (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝑥𝑦𝑦𝑥) → (¬ (𝐹𝑦) = (𝐹𝑥) ∨ ¬ (𝐹𝑥) = (𝐹𝑦))))
53 oridm 905 . . . . . . . . . . . . . . 15 ((¬ (𝐹𝑥) = (𝐹𝑦) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ¬ (𝐹𝑥) = (𝐹𝑦))
54 eqcom 2744 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
5554notbii 320 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑥) = (𝐹𝑦) ↔ ¬ (𝐹𝑦) = (𝐹𝑥))
5655orbi1i 914 . . . . . . . . . . . . . . 15 ((¬ (𝐹𝑥) = (𝐹𝑦) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ (¬ (𝐹𝑦) = (𝐹𝑥) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)))
5753, 56bitr3i 277 . . . . . . . . . . . . . 14 (¬ (𝐹𝑥) = (𝐹𝑦) ↔ (¬ (𝐹𝑦) = (𝐹𝑥) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)))
5852, 57imbitrrdi 252 . . . . . . . . . . . . 13 (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝑥𝑦𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
5958con2d 134 . . . . . . . . . . . 12 (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝐹𝑥) = (𝐹𝑦) → ¬ (𝑥𝑦𝑦𝑥)))
60 eloni 6394 . . . . . . . . . . . . 13 (𝑥 ∈ On → Ord 𝑥)
61 eloni 6394 . . . . . . . . . . . . 13 (𝑦 ∈ On → Ord 𝑦)
62 ordtri3 6420 . . . . . . . . . . . . . 14 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 = 𝑦 ↔ ¬ (𝑥𝑦𝑦𝑥)))
6362biimprd 248 . . . . . . . . . . . . 13 ((Ord 𝑥 ∧ Ord 𝑦) → (¬ (𝑥𝑦𝑦𝑥) → 𝑥 = 𝑦))
6460, 61, 63syl2an 596 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (¬ (𝑥𝑦𝑦𝑥) → 𝑥 = 𝑦))
6559, 64syl9r 78 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6651, 65syl6 35 . . . . . . . . . 10 (𝐴 ⊆ On → ((𝑥𝐴𝑦𝐴) → (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
6766imp32 418 . . . . . . . . 9 ((𝐴 ⊆ On ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6848, 67sylbid 240 . . . . . . . 8 ((𝐴 ⊆ On ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))
6968exp32 420 . . . . . . 7 (𝐴 ⊆ On → ((𝑥𝐴𝑦𝐴) → (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))))
7069a2d 29 . . . . . 6 (𝐴 ⊆ On → (((𝑥𝐴𝑦𝐴) → ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))) → ((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))))
71702alimdv 1918 . . . . 5 (𝐴 ⊆ On → (∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))) → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))))
72 r2al 3195 . . . . 5 (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))))
73 r2al 3195 . . . . 5 (∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
7471, 72, 733imtr4g 296 . . . 4 (𝐴 ⊆ On → (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
7544, 74syl5 34 . . 3 (𝐴 ⊆ On → (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
7675imdistani 568 . 2 ((𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
77 tz7.48.1 . . . 4 𝐹 Fn On
78 fnssres 6691 . . . 4 ((𝐹 Fn On ∧ 𝐴 ⊆ On) → (𝐹𝐴) Fn 𝐴)
7977, 78mpan 690 . . 3 (𝐴 ⊆ On → (𝐹𝐴) Fn 𝐴)
80 dffn2 6738 . . . 4 ((𝐹𝐴) Fn 𝐴 ↔ (𝐹𝐴):𝐴⟶V)
81 dff13 7275 . . . . . 6 ((𝐹𝐴):𝐴1-1→V ↔ ((𝐹𝐴):𝐴⟶V ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
82 df-f1 6566 . . . . . 6 ((𝐹𝐴):𝐴1-1→V ↔ ((𝐹𝐴):𝐴⟶V ∧ Fun (𝐹𝐴)))
8381, 82bitr3i 277 . . . . 5 (((𝐹𝐴):𝐴⟶V ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹𝐴):𝐴⟶V ∧ Fun (𝐹𝐴)))
8483simprbi 496 . . . 4 (((𝐹𝐴):𝐴⟶V ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun (𝐹𝐴))
8580, 84sylanb 581 . . 3 (((𝐹𝐴) Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun (𝐹𝐴))
8679, 85sylan 580 . 2 ((𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun (𝐹𝐴))
8776, 86syl 17 1 ((𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  wal 1538   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951  ccnv 5684  cres 5687  Ord word 6383  Oncon0 6384  Fun wfun 6555   Fn wfn 6556  wf 6557  1-1wf1 6558  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-ord 6387  df-on 6388  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569
This theorem is referenced by:  tz7.48-2  8482  tz7.49  8485  zorn2lem4  10539
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