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Theorem tz7.48lem 7819
 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 3120 . . . . . . 7 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
2 simpl 476 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐴) → 𝑥𝐴)
32anim1i 608 . . . . . . . . . 10 (((𝑥𝐴𝑦𝐴) ∧ 𝑦𝑥) → (𝑥𝐴𝑦𝑥))
43imim1i 63 . . . . . . . . 9 (((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)) → (((𝑥𝐴𝑦𝐴) ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
54expd 406 . . . . . . . 8 (((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)) → ((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
652alimi 1856 . . . . . . 7 (∀𝑥𝑦((𝑥𝐴𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)) → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
71, 6sylbi 209 . . . . . 6 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
8 r2al 3120 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
97, 8sylibr 226 . . . . 5 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
10 elequ1 2113 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑦𝑥𝑤𝑥))
11 fveq2 6446 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
1211eqeq2d 2787 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑥) = (𝐹𝑤)))
1312notbid 310 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (¬ (𝐹𝑥) = (𝐹𝑦) ↔ ¬ (𝐹𝑥) = (𝐹𝑤)))
1410, 13imbi12d 336 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤))))
1514cbvralv 3366 . . . . . . . . . 10 (∀𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)))
1615ralbii 3161 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑥𝐴𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)))
17 elequ2 2120 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑤𝑥𝑤𝑧))
18 fveqeq2 6455 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑥) = (𝐹𝑤) ↔ (𝐹𝑧) = (𝐹𝑤)))
1918notbid 310 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (¬ (𝐹𝑥) = (𝐹𝑤) ↔ ¬ (𝐹𝑧) = (𝐹𝑤)))
2017, 19imbi12d 336 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)) ↔ (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤))))
2120ralbidv 3167 . . . . . . . . . 10 (𝑥 = 𝑧 → (∀𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)) ↔ ∀𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤))))
2221cbvralv 3366 . . . . . . . . 9 (∀𝑥𝐴𝑤𝐴 (𝑤𝑥 → ¬ (𝐹𝑥) = (𝐹𝑤)) ↔ ∀𝑧𝐴𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)))
23 elequ1 2113 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
24 fveq2 6446 . . . . . . . . . . . . . . 15 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
2524eqeq2d 2787 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑧) = (𝐹𝑥)))
2625notbid 310 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (¬ (𝐹𝑧) = (𝐹𝑤) ↔ ¬ (𝐹𝑧) = (𝐹𝑥)))
2723, 26imbi12d 336 . . . . . . . . . . . 12 (𝑤 = 𝑥 → ((𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥))))
2827cbvralv 3366 . . . . . . . . . . 11 (∀𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ ∀𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)))
2928ralbii 3161 . . . . . . . . . 10 (∀𝑧𝐴𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ ∀𝑧𝐴𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)))
30 elequ2 2120 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
31 fveqeq2 6455 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → ((𝐹𝑧) = (𝐹𝑥) ↔ (𝐹𝑦) = (𝐹𝑥)))
3231notbid 310 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (¬ (𝐹𝑧) = (𝐹𝑥) ↔ ¬ (𝐹𝑦) = (𝐹𝑥)))
3330, 32imbi12d 336 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ((𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)) ↔ (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥))))
3433ralbidv 3167 . . . . . . . . . . 11 (𝑧 = 𝑦 → (∀𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)) ↔ ∀𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥))))
3534cbvralv 3366 . . . . . . . . . 10 (∀𝑧𝐴𝑥𝐴 (𝑥𝑧 → ¬ (𝐹𝑧) = (𝐹𝑥)) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
3629, 35bitri 267 . . . . . . . . 9 (∀𝑧𝐴𝑤𝐴 (𝑤𝑧 → ¬ (𝐹𝑧) = (𝐹𝑤)) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
3716, 22, 363bitri 289 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
38 ralcom2 3289 . . . . . . . 8 (∀𝑦𝐴𝑥𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
3937, 38sylbi 209 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)))
4039ancri 545 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ ∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
41 r19.26-2 3250 . . . . . 6 (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) ↔ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ ∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
4240, 41sylibr 226 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
439, 42syl 17 . . . 4 (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))
44 fvres 6465 . . . . . . . . . . 11 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
45 fvres 6465 . . . . . . . . . . 11 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
4644, 45eqeqan12d 2793 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
4746ad2antrl 718 . . . . . . . . 9 ((𝐴 ⊆ On ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
48 ssel 3814 . . . . . . . . . . . 12 (𝐴 ⊆ On → (𝑥𝐴𝑥 ∈ On))
49 ssel 3814 . . . . . . . . . . . 12 (𝐴 ⊆ On → (𝑦𝐴𝑦 ∈ On))
5048, 49anim12d 602 . . . . . . . . . . 11 (𝐴 ⊆ On → ((𝑥𝐴𝑦𝐴) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
51 pm3.48 949 . . . . . . . . . . . . . 14 (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝑥𝑦𝑦𝑥) → (¬ (𝐹𝑦) = (𝐹𝑥) ∨ ¬ (𝐹𝑥) = (𝐹𝑦))))
52 oridm 891 . . . . . . . . . . . . . . 15 ((¬ (𝐹𝑥) = (𝐹𝑦) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ ¬ (𝐹𝑥) = (𝐹𝑦))
53 eqcom 2784 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
5453notbii 312 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑥) = (𝐹𝑦) ↔ ¬ (𝐹𝑦) = (𝐹𝑥))
5554orbi1i 900 . . . . . . . . . . . . . . 15 ((¬ (𝐹𝑥) = (𝐹𝑦) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ (¬ (𝐹𝑦) = (𝐹𝑥) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)))
5652, 55bitr3i 269 . . . . . . . . . . . . . 14 (¬ (𝐹𝑥) = (𝐹𝑦) ↔ (¬ (𝐹𝑦) = (𝐹𝑥) ∨ ¬ (𝐹𝑥) = (𝐹𝑦)))
5751, 56syl6ibr 244 . . . . . . . . . . . . 13 (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝑥𝑦𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
5857con2d 132 . . . . . . . . . . . 12 (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝐹𝑥) = (𝐹𝑦) → ¬ (𝑥𝑦𝑦𝑥)))
59 eloni 5986 . . . . . . . . . . . . 13 (𝑥 ∈ On → Ord 𝑥)
60 eloni 5986 . . . . . . . . . . . . 13 (𝑦 ∈ On → Ord 𝑦)
61 ordtri3 6012 . . . . . . . . . . . . . 14 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 = 𝑦 ↔ ¬ (𝑥𝑦𝑦𝑥)))
6261biimprd 240 . . . . . . . . . . . . 13 ((Ord 𝑥 ∧ Ord 𝑦) → (¬ (𝑥𝑦𝑦𝑥) → 𝑥 = 𝑦))
6359, 60, 62syl2an 589 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (¬ (𝑥𝑦𝑦𝑥) → 𝑥 = 𝑦))
6458, 63syl9r 78 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6550, 64syl6 35 . . . . . . . . . 10 (𝐴 ⊆ On → ((𝑥𝐴𝑦𝐴) → (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
6665imp32 411 . . . . . . . . 9 ((𝐴 ⊆ On ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6747, 66sylbid 232 . . . . . . . 8 ((𝐴 ⊆ On ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))
6867exp32 413 . . . . . . 7 (𝐴 ⊆ On → ((𝑥𝐴𝑦𝐴) → (((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))))
6968a2d 29 . . . . . 6 (𝐴 ⊆ On → (((𝑥𝐴𝑦𝐴) → ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))) → ((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))))
70692alimdv 1961 . . . . 5 (𝐴 ⊆ On → (∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))) → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦))))
71 r2al 3120 . . . . 5 (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))))
72 r2al 3120 . . . . 5 (∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
7370, 71, 723imtr4g 288 . . . 4 (𝐴 ⊆ On → (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ¬ (𝐹𝑦) = (𝐹𝑥)) ∧ (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))) → ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
7443, 73syl5 34 . . 3 (𝐴 ⊆ On → (∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
7574imdistani 564 . 2 ((𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
76 tz7.48.1 . . . 4 𝐹 Fn On
77 fnssres 6250 . . . 4 ((𝐹 Fn On ∧ 𝐴 ⊆ On) → (𝐹𝐴) Fn 𝐴)
7876, 77mpan 680 . . 3 (𝐴 ⊆ On → (𝐹𝐴) Fn 𝐴)
79 dffn2 6293 . . . 4 ((𝐹𝐴) Fn 𝐴 ↔ (𝐹𝐴):𝐴⟶V)
80 dff13 6784 . . . . . 6 ((𝐹𝐴):𝐴1-1→V ↔ ((𝐹𝐴):𝐴⟶V ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)))
81 df-f1 6140 . . . . . 6 ((𝐹𝐴):𝐴1-1→V ↔ ((𝐹𝐴):𝐴⟶V ∧ Fun (𝐹𝐴)))
8280, 81bitr3i 269 . . . . 5 (((𝐹𝐴):𝐴⟶V ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹𝐴):𝐴⟶V ∧ Fun (𝐹𝐴)))
8382simprbi 492 . . . 4 (((𝐹𝐴):𝐴⟶V ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun (𝐹𝐴))
8479, 83sylanb 576 . . 3 (((𝐹𝐴) Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun (𝐹𝐴))
8578, 84sylan 575 . 2 ((𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝐴 (((𝐹𝐴)‘𝑥) = ((𝐹𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun (𝐹𝐴))
8675, 85syl 17 1 ((𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836  ∀wal 1599   = wceq 1601   ∈ wcel 2106  ∀wral 3089  Vcvv 3397   ⊆ wss 3791  ◡ccnv 5354   ↾ cres 5357  Ord word 5975  Oncon0 5976  Fun wfun 6129   Fn wfn 6130  ⟶wf 6131  –1-1→wf1 6132  ‘cfv 6135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-res 5367  df-ord 5979  df-on 5980  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fv 6143 This theorem is referenced by:  tz7.48-2  7820  tz7.49  7823  zorn2lem4  9656
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