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Theorem fofinf1o 8448
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 1166 . . . 4 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴onto𝐵)
2 fof 6298 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
31, 2syl 17 . . 3 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴𝐵)
4 domnsym 8293 . . . . . . 7 (𝐵 ≼ (𝐴 ∖ {𝑦}) → ¬ (𝐴 ∖ {𝑦}) ≺ 𝐵)
5 simp3 1168 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐵 ∈ Fin)
6 simp2 1167 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐴𝐵)
7 enfii 8384 . . . . . . . . . . 11 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)
85, 6, 7syl2anc 579 . . . . . . . . . 10 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐴 ∈ Fin)
98ad2antrr 717 . . . . . . . . 9 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴 ∈ Fin)
10 difssd 3900 . . . . . . . . . 10 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ⊆ 𝐴)
11 simplrr 796 . . . . . . . . . . . 12 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦𝐴)
12 neldifsn 4477 . . . . . . . . . . . 12 ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})
13 nelne1 3033 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
1411, 12, 13sylancl 580 . . . . . . . . . . 11 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
1514necomd 2992 . . . . . . . . . 10 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
16 df-pss 3748 . . . . . . . . . 10 ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ ((𝐴 ∖ {𝑦}) ⊆ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≠ 𝐴))
1710, 15, 16sylanbrc 578 . . . . . . . . 9 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
18 php3 8353 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊊ 𝐴) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
199, 17, 18syl2anc 579 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
206ad2antrr 717 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴𝐵)
21 sdomentr 8301 . . . . . . . 8 (((𝐴 ∖ {𝑦}) ≺ 𝐴𝐴𝐵) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
2219, 20, 21syl2anc 579 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
234, 22nsyl3 135 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 𝐵 ≼ (𝐴 ∖ {𝑦}))
248adantr 472 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐴 ∈ Fin)
25 difss 3899 . . . . . . . . . . 11 (𝐴 ∖ {𝑦}) ⊆ 𝐴
26 ssfi 8387 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐴 ∖ {𝑦}) ∈ Fin)
2724, 25, 26sylancl 580 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐴 ∖ {𝑦}) ∈ Fin)
283adantr 472 . . . . . . . . . . . 12 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐹:𝐴𝐵)
29 fssres 6252 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵 ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵)
3028, 25, 29sylancl 580 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵)
311adantr 472 . . . . . . . . . . . . . 14 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐹:𝐴onto𝐵)
32 foelrn 6568 . . . . . . . . . . . . . 14 ((𝐹:𝐴onto𝐵𝑧𝐵) → ∃𝑢𝐴 𝑧 = (𝐹𝑢))
3331, 32sylan 575 . . . . . . . . . . . . 13 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑧𝐵) → ∃𝑢𝐴 𝑧 = (𝐹𝑢))
34 simprll 797 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝑥𝐴)
35 simprrr 800 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝑥𝑦)
36 eldifsn 4472 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑥𝐴𝑥𝑦))
3734, 35, 36sylanbrc 578 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝑥 ∈ (𝐴 ∖ {𝑦}))
38 simprrl 799 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹𝑥) = (𝐹𝑦))
3938eqcomd 2771 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹𝑦) = (𝐹𝑥))
40 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
4140rspceeqv 3479 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹𝑦) = (𝐹𝑥)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤))
4237, 39, 41syl2anc 579 . . . . . . . . . . . . . . . . . . . 20 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤))
43 fveqeq2 6384 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑦 → ((𝐹𝑢) = (𝐹𝑤) ↔ (𝐹𝑦) = (𝐹𝑤)))
4443rexbidv 3199 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑦 → (∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤)))
4542, 44syl5ibrcom 238 . . . . . . . . . . . . . . . . . . 19 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
4645adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
4746imp 395 . . . . . . . . . . . . . . . . 17 (((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) ∧ 𝑢 = 𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
48 eldifsn 4472 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑢𝐴𝑢𝑦))
49 eqid 2765 . . . . . . . . . . . . . . . . . . . 20 (𝐹𝑢) = (𝐹𝑢)
50 fveq2 6375 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑢 → (𝐹𝑤) = (𝐹𝑢))
5150rspceeqv 3479 . . . . . . . . . . . . . . . . . . . 20 ((𝑢 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹𝑢) = (𝐹𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5249, 51mpan2 682 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴 ∖ {𝑦}) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5348, 52sylbir 226 . . . . . . . . . . . . . . . . . 18 ((𝑢𝐴𝑢𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5453adantll 705 . . . . . . . . . . . . . . . . 17 (((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) ∧ 𝑢𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5547, 54pm2.61dane 3024 . . . . . . . . . . . . . . . 16 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
56 fvres 6394 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝐴 ∖ {𝑦}) → ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) = (𝐹𝑤))
5756eqeq2d 2775 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝐴 ∖ {𝑦}) → (𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ 𝑧 = (𝐹𝑤)))
5857rexbiia 3187 . . . . . . . . . . . . . . . . 17 (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹𝑤))
59 eqeq1 2769 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝐹𝑢) → (𝑧 = (𝐹𝑤) ↔ (𝐹𝑢) = (𝐹𝑤)))
6059rexbidv 3199 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐹𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
6158, 60syl5bb 274 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
6255, 61syl5ibrcom 238 . . . . . . . . . . . . . . 15 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → (𝑧 = (𝐹𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
6362rexlimdva 3178 . . . . . . . . . . . . . 14 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (∃𝑢𝐴 𝑧 = (𝐹𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
6463imp 395 . . . . . . . . . . . . 13 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ ∃𝑢𝐴 𝑧 = (𝐹𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
6533, 64syldan 585 . . . . . . . . . . . 12 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑧𝐵) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
6665ralrimiva 3113 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → ∀𝑧𝐵𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
67 dffo3 6564 . . . . . . . . . . 11 ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵 ↔ ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵 ∧ ∀𝑧𝐵𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
6830, 66, 67sylanbrc 578 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵)
69 fodomfi 8446 . . . . . . . . . 10 (((𝐴 ∖ {𝑦}) ∈ Fin ∧ (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7027, 68, 69syl2anc 579 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7170anassrs 459 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7271expr 448 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥𝑦𝐵 ≼ (𝐴 ∖ {𝑦})))
7372necon1bd 2955 . . . . . 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 401 . . . 4 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
7675ralrimivva 3118 . . 3 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
77 dff13 6704 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
783, 76, 77sylanbrc 578 . 2 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1𝐵)
79 df-f1o 6075 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
8078, 1, 79sylanbrc 578 1 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  cdif 3729  wss 3732  wpss 3733  {csn 4334   class class class wbr 4809  cres 5279  wf 6064  1-1wf1 6065  ontowfo 6066  1-1-ontowf1o 6067  cfv 6068  cen 8157  cdom 8158  csdm 8159  Fincfn 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-1o 7764  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164
This theorem is referenced by:  rneqdmfinf1o  8449  phpreu  33749
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