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Theorem fofinf1o 8406
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 1131 . . . 4 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴onto𝐵)
2 fof 6276 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
31, 2syl 17 . . 3 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴𝐵)
4 domnsym 8251 . . . . . . 7 (𝐵 ≼ (𝐴 ∖ {𝑦}) → ¬ (𝐴 ∖ {𝑦}) ≺ 𝐵)
5 simp3 1133 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐵 ∈ Fin)
6 simp2 1132 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐴𝐵)
7 enfii 8342 . . . . . . . . . . 11 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)
85, 6, 7syl2anc 696 . . . . . . . . . 10 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐴 ∈ Fin)
98ad2antrr 764 . . . . . . . . 9 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴 ∈ Fin)
10 difssd 3881 . . . . . . . . . 10 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ⊆ 𝐴)
11 simplrr 820 . . . . . . . . . . . 12 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦𝐴)
12 neldifsn 4467 . . . . . . . . . . . 12 ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})
13 nelne1 3028 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
1411, 12, 13sylancl 697 . . . . . . . . . . 11 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
1514necomd 2987 . . . . . . . . . 10 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
16 df-pss 3731 . . . . . . . . . 10 ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ ((𝐴 ∖ {𝑦}) ⊆ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≠ 𝐴))
1710, 15, 16sylanbrc 701 . . . . . . . . 9 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
18 php3 8311 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊊ 𝐴) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
199, 17, 18syl2anc 696 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
206ad2antrr 764 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴𝐵)
21 sdomentr 8259 . . . . . . . 8 (((𝐴 ∖ {𝑦}) ≺ 𝐴𝐴𝐵) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
2219, 20, 21syl2anc 696 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
234, 22nsyl3 133 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 𝐵 ≼ (𝐴 ∖ {𝑦}))
248adantr 472 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐴 ∈ Fin)
25 difss 3880 . . . . . . . . . . 11 (𝐴 ∖ {𝑦}) ⊆ 𝐴
26 ssfi 8345 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐴 ∖ {𝑦}) ∈ Fin)
2724, 25, 26sylancl 697 . . . . . . . . . 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 6231 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵 ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵)
3028, 25, 29sylancl 697 . . . . . . . . . . 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 6541 . . . . . . . . . . . . . 14 ((𝐹:𝐴onto𝐵𝑧𝐵) → ∃𝑢𝐴 𝑧 = (𝐹𝑢))
3331, 32sylan 489 . . . . . . . . . . . . 13 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑧𝐵) → ∃𝑢𝐴 𝑧 = (𝐹𝑢))
34 simprll 821 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝑥𝐴)
35 simprrr 824 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝑥𝑦)
36 eldifsn 4462 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑥𝐴𝑥𝑦))
3734, 35, 36sylanbrc 701 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝑥 ∈ (𝐴 ∖ {𝑦}))
38 simprrl 823 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹𝑥) = (𝐹𝑦))
3938eqcomd 2766 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹𝑦) = (𝐹𝑥))
40 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
4140eqeq2d 2770 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑥 → ((𝐹𝑦) = (𝐹𝑤) ↔ (𝐹𝑦) = (𝐹𝑥)))
4241rspcev 3449 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹𝑦) = (𝐹𝑥)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤))
4337, 39, 42syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤))
44 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑦 → (𝐹𝑢) = (𝐹𝑦))
4544eqeq1d 2762 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑦 → ((𝐹𝑢) = (𝐹𝑤) ↔ (𝐹𝑦) = (𝐹𝑤)))
4645rexbidv 3190 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑦 → (∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤)))
4743, 46syl5ibrcom 237 . . . . . . . . . . . . . . . . . . 19 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
4847adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
4948imp 444 . . . . . . . . . . . . . . . . 17 (((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) ∧ 𝑢 = 𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
50 eldifsn 4462 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑢𝐴𝑢𝑦))
51 eqid 2760 . . . . . . . . . . . . . . . . . . . 20 (𝐹𝑢) = (𝐹𝑢)
52 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑢 → (𝐹𝑤) = (𝐹𝑢))
5352eqeq2d 2770 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑢 → ((𝐹𝑢) = (𝐹𝑤) ↔ (𝐹𝑢) = (𝐹𝑢)))
5453rspcev 3449 . . . . . . . . . . . . . . . . . . . 20 ((𝑢 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹𝑢) = (𝐹𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5551, 54mpan2 709 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴 ∖ {𝑦}) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5650, 55sylbir 225 . . . . . . . . . . . . . . . . . 18 ((𝑢𝐴𝑢𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5756adantll 752 . . . . . . . . . . . . . . . . 17 (((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) ∧ 𝑢𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5849, 57pm2.61dane 3019 . . . . . . . . . . . . . . . 16 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
59 fvres 6368 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝐴 ∖ {𝑦}) → ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) = (𝐹𝑤))
6059eqeq2d 2770 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝐴 ∖ {𝑦}) → (𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ 𝑧 = (𝐹𝑤)))
6160rexbiia 3178 . . . . . . . . . . . . . . . . 17 (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹𝑤))
62 eqeq1 2764 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝐹𝑢) → (𝑧 = (𝐹𝑤) ↔ (𝐹𝑢) = (𝐹𝑤)))
6362rexbidv 3190 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐹𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
6461, 63syl5bb 272 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
6558, 64syl5ibrcom 237 . . . . . . . . . . . . . . 15 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → (𝑧 = (𝐹𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
6665rexlimdva 3169 . . . . . . . . . . . . . 14 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (∃𝑢𝐴 𝑧 = (𝐹𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
6766imp 444 . . . . . . . . . . . . 13 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ ∃𝑢𝐴 𝑧 = (𝐹𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
6833, 67syldan 488 . . . . . . . . . . . 12 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑧𝐵) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
6968ralrimiva 3104 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → ∀𝑧𝐵𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
70 dffo3 6537 . . . . . . . . . . 11 ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵 ↔ ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵 ∧ ∀𝑧𝐵𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
7130, 69, 70sylanbrc 701 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵)
72 fodomfi 8404 . . . . . . . . . 10 (((𝐴 ∖ {𝑦}) ∈ Fin ∧ (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7327, 71, 72syl2anc 696 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7473anassrs 683 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7574expr 644 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥𝑦𝐵 ≼ (𝐴 ∖ {𝑦})))
7675necon1bd 2950 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (¬ 𝐵 ≼ (𝐴 ∖ {𝑦}) → 𝑥 = 𝑦))
7723, 76mpd 15 . . . . 5 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 = 𝑦)
7877ex 449 . . . 4 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
7978ralrimivva 3109 . . 3 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
80 dff13 6675 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
813, 79, 80sylanbrc 701 . 2 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1𝐵)
82 df-f1o 6056 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
8381, 1, 82sylanbrc 701 1 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  cdif 3712  wss 3715  wpss 3716  {csn 4321   class class class wbr 4804  cres 5268  wf 6045  1-1wf1 6046  ontowfo 6047  1-1-ontowf1o 6048  cfv 6049  cen 8118  cdom 8119  csdm 8120  Fincfn 8121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-1o 7729  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125
This theorem is referenced by:  rneqdmfinf1o  8407  phpreu  33706
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