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Theorem fofinf1o 9329
Description: Any surjection from one finite set to another of equal size must be a bijection. (Contributed by Mario Carneiro, 19-Aug-2014.)
Assertion
Ref Expression
fofinf1o ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1-onto𝐵)

Proof of Theorem fofinf1o
Dummy variables 𝑤 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1134 . . . 4 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴onto𝐵)
2 fof 6804 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
31, 2syl 17 . . 3 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴𝐵)
4 domnsym 9101 . . . . . . 7 (𝐵 ≼ (𝐴 ∖ {𝑦}) → ¬ (𝐴 ∖ {𝑦}) ≺ 𝐵)
5 simp3 1136 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐵 ∈ Fin)
6 simp2 1135 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐴𝐵)
7 enfii 9191 . . . . . . . . . . 11 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)
85, 6, 7syl2anc 582 . . . . . . . . . 10 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐴 ∈ Fin)
98ad2antrr 722 . . . . . . . . 9 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴 ∈ Fin)
10 difssd 4131 . . . . . . . . . 10 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ⊆ 𝐴)
11 simplrr 774 . . . . . . . . . . . 12 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦𝐴)
12 neldifsn 4794 . . . . . . . . . . . 12 ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})
13 nelne1 3037 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
1411, 12, 13sylancl 584 . . . . . . . . . . 11 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
1514necomd 2994 . . . . . . . . . 10 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
16 df-pss 3966 . . . . . . . . . 10 ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ ((𝐴 ∖ {𝑦}) ⊆ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≠ 𝐴))
1710, 15, 16sylanbrc 581 . . . . . . . . 9 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
18 php3 9214 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊊ 𝐴) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
199, 17, 18syl2anc 582 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
206ad2antrr 722 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴𝐵)
21 sdomentr 9113 . . . . . . . 8 (((𝐴 ∖ {𝑦}) ≺ 𝐴𝐴𝐵) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
2219, 20, 21syl2anc 582 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
234, 22nsyl3 138 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 𝐵 ≼ (𝐴 ∖ {𝑦}))
248adantr 479 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐴 ∈ Fin)
25 difss 4130 . . . . . . . . . . 11 (𝐴 ∖ {𝑦}) ⊆ 𝐴
26 ssfi 9175 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐴 ∖ {𝑦}) ∈ Fin)
2724, 25, 26sylancl 584 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐴 ∖ {𝑦}) ∈ Fin)
283adantr 479 . . . . . . . . . . . 12 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐹:𝐴𝐵)
29 fssres 6756 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵 ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵)
3028, 25, 29sylancl 584 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵)
311adantr 479 . . . . . . . . . . . . . 14 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐹:𝐴onto𝐵)
32 foelrn 7107 . . . . . . . . . . . . . 14 ((𝐹:𝐴onto𝐵𝑧𝐵) → ∃𝑢𝐴 𝑧 = (𝐹𝑢))
3331, 32sylan 578 . . . . . . . . . . . . 13 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑧𝐵) → ∃𝑢𝐴 𝑧 = (𝐹𝑢))
34 simprll 775 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝑥𝐴)
35 simprrr 778 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝑥𝑦)
36 eldifsn 4789 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑥𝐴𝑥𝑦))
3734, 35, 36sylanbrc 581 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝑥 ∈ (𝐴 ∖ {𝑦}))
38 simprrl 777 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹𝑥) = (𝐹𝑦))
3938eqcomd 2736 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹𝑦) = (𝐹𝑥))
40 fveq2 6890 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
4140rspceeqv 3632 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹𝑦) = (𝐹𝑥)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤))
4237, 39, 41syl2anc 582 . . . . . . . . . . . . . . . . . . . 20 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤))
43 fveqeq2 6899 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑦 → ((𝐹𝑢) = (𝐹𝑤) ↔ (𝐹𝑦) = (𝐹𝑤)))
4443rexbidv 3176 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑦 → (∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤)))
4542, 44syl5ibrcom 246 . . . . . . . . . . . . . . . . . . 19 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
4645adantr 479 . . . . . . . . . . . . . . . . . 18 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
4746imp 405 . . . . . . . . . . . . . . . . 17 (((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) ∧ 𝑢 = 𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
48 eldifsn 4789 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑢𝐴𝑢𝑦))
49 eqid 2730 . . . . . . . . . . . . . . . . . . . 20 (𝐹𝑢) = (𝐹𝑢)
50 fveq2 6890 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑢 → (𝐹𝑤) = (𝐹𝑢))
5150rspceeqv 3632 . . . . . . . . . . . . . . . . . . . 20 ((𝑢 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹𝑢) = (𝐹𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5249, 51mpan2 687 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴 ∖ {𝑦}) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5348, 52sylbir 234 . . . . . . . . . . . . . . . . . 18 ((𝑢𝐴𝑢𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5453adantll 710 . . . . . . . . . . . . . . . . 17 (((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) ∧ 𝑢𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5547, 54pm2.61dane 3027 . . . . . . . . . . . . . . . 16 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
56 fvres 6909 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝐴 ∖ {𝑦}) → ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) = (𝐹𝑤))
5756eqeq2d 2741 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝐴 ∖ {𝑦}) → (𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ 𝑧 = (𝐹𝑤)))
5857rexbiia 3090 . . . . . . . . . . . . . . . . 17 (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹𝑤))
59 eqeq1 2734 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝐹𝑢) → (𝑧 = (𝐹𝑤) ↔ (𝐹𝑢) = (𝐹𝑤)))
6059rexbidv 3176 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐹𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
6158, 60bitrid 282 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
6255, 61syl5ibrcom 246 . . . . . . . . . . . . . . 15 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → (𝑧 = (𝐹𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
6362rexlimdva 3153 . . . . . . . . . . . . . 14 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (∃𝑢𝐴 𝑧 = (𝐹𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
6463imp 405 . . . . . . . . . . . . 13 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ ∃𝑢𝐴 𝑧 = (𝐹𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
6533, 64syldan 589 . . . . . . . . . . . 12 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑧𝐵) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
6665ralrimiva 3144 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → ∀𝑧𝐵𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
67 dffo3 7102 . . . . . . . . . . 11 ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵 ↔ ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵 ∧ ∀𝑧𝐵𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
6830, 66, 67sylanbrc 581 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵)
69 fodomfi 9327 . . . . . . . . . 10 (((𝐴 ∖ {𝑦}) ∈ Fin ∧ (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7027, 68, 69syl2anc 582 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7170anassrs 466 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7271expr 455 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥𝑦𝐵 ≼ (𝐴 ∖ {𝑦})))
7372necon1bd 2956 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (¬ 𝐵 ≼ (𝐴 ∖ {𝑦}) → 𝑥 = 𝑦))
7423, 73mpd 15 . . . . 5 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 = 𝑦)
7574ex 411 . . . 4 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
7675ralrimivva 3198 . . 3 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
77 dff13 7256 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
783, 76, 77sylanbrc 581 . 2 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1𝐵)
79 df-f1o 6549 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
8078, 1, 79sylanbrc 581 1 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1085   = wceq 1539  wcel 2104  wne 2938  wral 3059  wrex 3068  cdif 3944  wss 3947  wpss 3948  {csn 4627   class class class wbr 5147  cres 5677  wf 6538  1-1wf1 6539  ontowfo 6540  1-1-ontowf1o 6541  cfv 6542  cen 8938  cdom 8939  csdm 8940  Fincfn 8941
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945
This theorem is referenced by:  rneqdmfinf1o  9330  phpreu  36775
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