MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fofinf1o Structured version   Visualization version   GIF version

Theorem fofinf1o 8186
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 1059 . . . 4 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴onto𝐵)
2 fof 6074 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
31, 2syl 17 . . 3 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴𝐵)
4 domnsym 8031 . . . . . . 7 (𝐵 ≼ (𝐴 ∖ {𝑦}) → ¬ (𝐴 ∖ {𝑦}) ≺ 𝐵)
5 simp3 1061 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐵 ∈ Fin)
6 simp2 1060 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐴𝐵)
7 enfii 8122 . . . . . . . . . . 11 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)
85, 6, 7syl2anc 692 . . . . . . . . . 10 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐴 ∈ Fin)
98ad2antrr 761 . . . . . . . . 9 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴 ∈ Fin)
10 difssd 3721 . . . . . . . . . 10 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ⊆ 𝐴)
11 simplrr 800 . . . . . . . . . . . 12 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦𝐴)
12 neldifsn 4295 . . . . . . . . . . . 12 ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})
13 nelne1 2892 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
1411, 12, 13sylancl 693 . . . . . . . . . . 11 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
1514necomd 2851 . . . . . . . . . 10 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
16 df-pss 3576 . . . . . . . . . 10 ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ ((𝐴 ∖ {𝑦}) ⊆ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≠ 𝐴))
1710, 15, 16sylanbrc 697 . . . . . . . . 9 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
18 php3 8091 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊊ 𝐴) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
199, 17, 18syl2anc 692 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
206ad2antrr 761 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝐴𝐵)
21 sdomentr 8039 . . . . . . . 8 (((𝐴 ∖ {𝑦}) ≺ 𝐴𝐴𝐵) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
2219, 20, 21syl2anc 692 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
234, 22nsyl3 133 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 𝐵 ≼ (𝐴 ∖ {𝑦}))
248adantr 481 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐴 ∈ Fin)
25 difss 3720 . . . . . . . . . . 11 (𝐴 ∖ {𝑦}) ⊆ 𝐴
26 ssfi 8125 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐴 ∖ {𝑦}) ∈ Fin)
2724, 25, 26sylancl 693 . . . . . . . . . 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 6029 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵 ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵)
3028, 25, 29sylancl 693 . . . . . . . . . . 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 6335 . . . . . . . . . . . . . 14 ((𝐹:𝐴onto𝐵𝑧𝐵) → ∃𝑢𝐴 𝑧 = (𝐹𝑢))
3331, 32sylan 488 . . . . . . . . . . . . 13 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑧𝐵) → ∃𝑢𝐴 𝑧 = (𝐹𝑢))
34 simprll 801 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝑥𝐴)
35 simprrr 804 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝑥𝑦)
36 eldifsn 4292 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑥𝐴𝑥𝑦))
3734, 35, 36sylanbrc 697 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝑥 ∈ (𝐴 ∖ {𝑦}))
38 simprrl 803 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹𝑥) = (𝐹𝑦))
3938eqcomd 2632 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹𝑦) = (𝐹𝑥))
40 fveq2 6150 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
4140eqeq2d 2636 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑥 → ((𝐹𝑦) = (𝐹𝑤) ↔ (𝐹𝑦) = (𝐹𝑥)))
4241rspcev 3300 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹𝑦) = (𝐹𝑥)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤))
4337, 39, 42syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑦) = (𝐹𝑤))
44 fveq2 6150 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑦 → (𝐹𝑢) = (𝐹𝑦))
4544eqeq1d 2628 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑦 → ((𝐹𝑢) = (𝐹𝑤) ↔ (𝐹𝑦) = (𝐹𝑤)))
4645rexbidv 3050 . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 18 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤)))
4948imp 445 . . . . . . . . . . . . . . . . 17 (((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) ∧ 𝑢 = 𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
50 eldifsn 4292 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑢𝐴𝑢𝑦))
51 eqid 2626 . . . . . . . . . . . . . . . . . . . 20 (𝐹𝑢) = (𝐹𝑢)
52 fveq2 6150 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑢 → (𝐹𝑤) = (𝐹𝑢))
5352eqeq2d 2636 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑢 → ((𝐹𝑢) = (𝐹𝑤) ↔ (𝐹𝑢) = (𝐹𝑢)))
5453rspcev 3300 . . . . . . . . . . . . . . . . . . . 20 ((𝑢 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹𝑢) = (𝐹𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5551, 54mpan2 706 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴 ∖ {𝑦}) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5650, 55sylbir 225 . . . . . . . . . . . . . . . . . 18 ((𝑢𝐴𝑢𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5756adantll 749 . . . . . . . . . . . . . . . . 17 (((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) ∧ 𝑢𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
5849, 57pm2.61dane 2883 . . . . . . . . . . . . . . . 16 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑢𝐴) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹𝑢) = (𝐹𝑤))
59 fvres 6165 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝐴 ∖ {𝑦}) → ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) = (𝐹𝑤))
6059eqeq2d 2636 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝐴 ∖ {𝑦}) → (𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ 𝑧 = (𝐹𝑤)))
6160rexbiia 3038 . . . . . . . . . . . . . . . . 17 (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹𝑤))
62 eqeq1 2630 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝐹𝑢) → (𝑧 = (𝐹𝑤) ↔ (𝐹𝑢) = (𝐹𝑤)))
6362rexbidv 3050 . . . . . . . . . . . . . . . . 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 3029 . . . . . . . . . . . . . 14 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (∃𝑢𝐴 𝑧 = (𝐹𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
6766imp 445 . . . . . . . . . . . . 13 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ ∃𝑢𝐴 𝑧 = (𝐹𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
6833, 67syldan 487 . . . . . . . . . . . 12 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) ∧ 𝑧𝐵) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
6968ralrimiva 2965 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → ∀𝑧𝐵𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
70 dffo3 6331 . . . . . . . . . . 11 ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵 ↔ ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵 ∧ ∀𝑧𝐵𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
7130, 69, 70sylanbrc 697 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵)
72 fodomfi 8184 . . . . . . . . . 10 (((𝐴 ∖ {𝑦}) ∈ Fin ∧ (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto𝐵) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7327, 71, 72syl2anc 692 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7473anassrs 679 . . . . . . . 8 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
7574expr 642 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥𝑦𝐵 ≼ (𝐴 ∖ {𝑦})))
7675necon1bd 2814 . . . . . 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 450 . . . 4 (((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
7978ralrimivva 2970 . . 3 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
80 dff13 6467 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
813, 79, 80sylanbrc 697 . 2 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1𝐵)
82 df-f1o 5857 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
8381, 1, 82sylanbrc 697 1 ((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wral 2912  wrex 2913  cdif 3557  wss 3560  wpss 3561  {csn 4153   class class class wbr 4618  cres 5081  wf 5846  1-1wf1 5847  ontowfo 5848  1-1-ontowf1o 5849  cfv 5850  cen 7897  cdom 7898  csdm 7899  Fincfn 7900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-1o 7506  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904
This theorem is referenced by:  rneqdmfinf1o  8187  phpreu  33011
  Copyright terms: Public domain W3C validator