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Theorem fofinf1o 8788
 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 1130 . . . 4 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴onto𝐵)
2 fof 6587 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
31, 2syl 17 . . 3 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴𝐵)
4 domnsym 8632 . . . . . . 7 (𝐵 ≼ (𝐴 ∖ {𝑦}) → ¬ (𝐴 ∖ {𝑦}) ≺ 𝐵)
5 simp3 1132 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐵 ∈ Fin)
6 simp2 1131 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐴𝐵)
7 enfii 8724 . . . . . . . . . . 11 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)
85, 6, 7syl2anc 584 . . . . . . . . . 10 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐴 ∈ Fin)
98ad2antrr 722 . . . . . . . . 9 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴 ∈ Fin)
10 difssd 4113 . . . . . . . . . 10 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ⊆ 𝐴)
11 simplrr 774 . . . . . . . . . . . 12 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦𝐴)
12 neldifsn 4724 . . . . . . . . . . . 12 ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})
13 nelne1 3118 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
1411, 12, 13sylancl 586 . . . . . . . . . . 11 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
1514necomd 3076 . . . . . . . . . 10 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
16 df-pss 3958 . . . . . . . . . 10 ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ ((𝐴 ∖ {𝑦}) ⊆ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≠ 𝐴))
1710, 15, 16sylanbrc 583 . . . . . . . . 9 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
18 php3 8692 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊊ 𝐴) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
199, 17, 18syl2anc 584 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
206ad2antrr 722 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴𝐵)
21 sdomentr 8640 . . . . . . . 8 (((𝐴 ∖ {𝑦}) ≺ 𝐴𝐴𝐵) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
2219, 20, 21syl2anc 584 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
234, 22nsyl3 140 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 𝐵 ≼ (𝐴 ∖ {𝑦}))
248adantr 481 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐴 ∈ Fin)
25 difss 4112 . . . . . . . . . . 11 (𝐴 ∖ {𝑦}) ⊆ 𝐴
26 ssfi 8727 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐴 ∖ {𝑦}) ∈ Fin)
2724, 25, 26sylancl 586 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐴 ∖ {𝑦}) ∈ Fin)
283adantr 481 . . . . . . . . . . . 12 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐹:𝐴𝐵)
29 fssres 6541 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵 ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵)
3028, 25, 29sylancl 586 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵)
311adantr 481 . . . . . . . . . . . . . 14 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐹:𝐴onto𝐵)
32 foelrn 6868 . . . . . . . . . . . . . 14 ((𝐹:𝐴onto𝐵𝑧𝐵) → ∃𝑢𝐴 𝑧 = (𝐹𝑢))
3331, 32sylan 580 . . . . . . . . . . . . 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 4718 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑥𝐴𝑥𝑦))
3734, 35, 36sylanbrc 583 . . . . . . . . . . . . . . . . . . . . 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 2832 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹𝑦) = (𝐹𝑥))
40 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
4140rspceeqv 3642 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹𝑦) = (𝐹𝑥)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤))
4237, 39, 41syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤))
43 fveqeq2 6676 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑦 → ((𝐹𝑢) = (𝐹𝑤) ↔ (𝐹𝑦) = (𝐹𝑤)))
4443rexbidv 3302 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑦 → (∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤)))
4542, 44syl5ibrcom 248 . . . . . . . . . . . . . . . . . . 19 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
4645adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
4746imp 407 . . . . . . . . . . . . . . . . 17 (((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) ∧ 𝑢 = 𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
48 eldifsn 4718 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑢𝐴𝑢𝑦))
49 eqid 2826 . . . . . . . . . . . . . . . . . . . 20 (𝐹𝑢) = (𝐹𝑢)
50 fveq2 6667 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑢 → (𝐹𝑤) = (𝐹𝑢))
5150rspceeqv 3642 . . . . . . . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . . . . . . 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 3109 . . . . . . . . . . . . . . . 16 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
56 fvres 6686 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝐴 ∖ {𝑦}) → ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) = (𝐹𝑤))
5756eqeq2d 2837 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝐴 ∖ {𝑦}) → (𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ 𝑧 = (𝐹𝑤)))
5857rexbiia 3251 . . . . . . . . . . . . . . . . 17 (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹𝑤))
59 eqeq1 2830 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝐹𝑢) → (𝑧 = (𝐹𝑤) ↔ (𝐹𝑢) = (𝐹𝑤)))
6059rexbidv 3302 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐹𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
6158, 60syl5bb 284 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
6255, 61syl5ibrcom 248 . . . . . . . . . . . . . . 15 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → (𝑧 = (𝐹𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
6362rexlimdva 3289 . . . . . . . . . . . . . 14 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (∃𝑢𝐴 𝑧 = (𝐹𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
6463imp 407 . . . . . . . . . . . . 13 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ ∃𝑢𝐴 𝑧 = (𝐹𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
6533, 64syldan 591 . . . . . . . . . . . 12 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑧𝐵) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
6665ralrimiva 3187 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → ∀𝑧𝐵𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
67 dffo3 6864 . . . . . . . . . . 11 ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵 ↔ ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵 ∧ ∀𝑧𝐵𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
6830, 66, 67sylanbrc 583 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵)
69 fodomfi 8786 . . . . . . . . . 10 (((𝐴 ∖ {𝑦}) ∈ Fin ∧ (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7027, 68, 69syl2anc 584 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7170anassrs 468 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7271expr 457 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥𝑦𝐵 ≼ (𝐴 ∖ {𝑦})))
7372necon1bd 3039 . . . . . 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 413 . . . 4 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
7675ralrimivva 3196 . . 3 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
77 dff13 7007 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
783, 76, 77sylanbrc 583 . 2 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1𝐵)
79 df-f1o 6359 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
8078, 1, 79sylanbrc 583 1 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1-onto𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   ≠ wne 3021  ∀wral 3143  ∃wrex 3144   ∖ cdif 3937   ⊆ wss 3940   ⊊ wpss 3941  {csn 4564   class class class wbr 5063   ↾ cres 5556  ⟶wf 6348  –1-1→wf1 6349  –onto→wfo 6350  –1-1-onto→wf1o 6351  ‘cfv 6352   ≈ cen 8495   ≼ cdom 8496   ≺ csdm 8497  Fincfn 8498 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-om 7569  df-1o 8093  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502 This theorem is referenced by:  rneqdmfinf1o  8789  phpreu  34743
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