Theorem fofinf1o 9310

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:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴–1-1-onto→𝐵)

Proof of Theorem fofinf1o

Dummy variables 𝑤 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴–onto→𝐵)
2		fof 6792	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	1, 2	syl 17	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴⟶𝐵)
4		domnsym 9082	. . . . . . 7 ⊢ (𝐵 ≼ (𝐴 ∖ {𝑦}) → ¬ (𝐴 ∖ {𝑦}) ≺ 𝐵)
5		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐵 ∈ Fin)
6		simp2 1137	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ≈ 𝐵)
7		enfii 9172	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
8	5, 6, 7	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin)
9	8	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐴 ∈ Fin)
10		difssd 4128	. . . . . . . . . 10 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ⊆ 𝐴)
11		simplrr 776	. . . . . . . . . . . 12 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ 𝐴)
12		neldifsn 4788	. . . . . . . . . . . 12 ⊢ ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})
13		nelne1 3038	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
14	11, 12, 13	sylancl 586	. . . . . . . . . . 11 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
15	14	necomd 2995	. . . . . . . . . 10 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
16		df-pss 3963	. . . . . . . . . 10 ⊢ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ ((𝐴 ∖ {𝑦}) ⊆ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≠ 𝐴))
17	10, 15, 16	sylanbrc 583	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
18		php3 9195	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊊ 𝐴) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
19	9, 17, 18	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
20	6	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐴 ≈ 𝐵)
21		sdomentr 9094	. . . . . . . 8 ⊢ (((𝐴 ∖ {𝑦}) ≺ 𝐴 ∧ 𝐴 ≈ 𝐵) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
22	19, 20, 21	syl2anc 584	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
23	4, 22	nsyl3 138	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → ¬ 𝐵 ≼ (𝐴 ∖ {𝑦}))
24	8	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝐴 ∈ Fin)
25		difss 4127	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝑦}) ⊆ 𝐴
26		ssfi 9156	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐴 ∖ {𝑦}) ∈ Fin)
27	24, 25, 26	sylancl 586	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (𝐴 ∖ {𝑦}) ∈ Fin)
28	3	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝐹:𝐴⟶𝐵)
29		fssres 6744	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵)
30	28, 25, 29	sylancl 586	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵)
31	1	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝐹:𝐴–onto→𝐵)
32		foelrn 7092	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑧 ∈ 𝐵) → ∃𝑢 ∈ 𝐴 𝑧 = (𝐹‘𝑢))
33	31, 32	sylan 580	. . . . . . . . . . . . 13 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑧 ∈ 𝐵) → ∃𝑢 ∈ 𝐴 𝑧 = (𝐹‘𝑢))
34		simprll 777	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝑥 ∈ 𝐴)
35		simprrr 780	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝑥 ≠ 𝑦)
36		eldifsn 4783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝑦))
37	34, 35, 36	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝑥 ∈ (𝐴 ∖ {𝑦}))
38		simprrl 779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
39	38	eqcomd 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (𝐹‘𝑦) = (𝐹‘𝑥))
40		fveq2 6878	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
41	40	rspceeqv 3629	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹‘𝑦) = (𝐹‘𝑥)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑦) = (𝐹‘𝑤))
42	37, 39, 41	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑦) = (𝐹‘𝑤))
43		fveqeq2 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑦 → ((𝐹‘𝑢) = (𝐹‘𝑤) ↔ (𝐹‘𝑦) = (𝐹‘𝑤)))
44	43	rexbidv 3177	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑦 → (∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑦) = (𝐹‘𝑤)))
45	42, 44	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤)))
46	45	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑢 ∈ 𝐴) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤)))
47	46	imp 407	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑢 ∈ 𝐴) ∧ 𝑢 = 𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))
48		eldifsn 4783	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑢 ∈ 𝐴 ∧ 𝑢 ≠ 𝑦))
49		eqid 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹‘𝑢) = (𝐹‘𝑢)
50		fveq2 6878	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑢 → (𝐹‘𝑤) = (𝐹‘𝑢))
51	50	rspceeqv 3629	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑢 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹‘𝑢) = (𝐹‘𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))
52	49, 51	mpan2 689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ (𝐴 ∖ {𝑦}) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))
53	48, 52	sylbir 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑢 ≠ 𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))
54	53	adantll 712	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑢 ∈ 𝐴) ∧ 𝑢 ≠ 𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))
55	47, 54	pm2.61dane 3028	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑢 ∈ 𝐴) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))
56		fvres 6897	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (𝐴 ∖ {𝑦}) → ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) = (𝐹‘𝑤))
57	56	eqeq2d 2742	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝐴 ∖ {𝑦}) → (𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ 𝑧 = (𝐹‘𝑤)))
58	57	rexbiia 3091	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹‘𝑤))
59		eqeq1 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐹‘𝑢) → (𝑧 = (𝐹‘𝑤) ↔ (𝐹‘𝑢) = (𝐹‘𝑤)))
60	59	rexbidv 3177	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐹‘𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤)))
61	58, 60	bitrid 282	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐹‘𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤)))
62	55, 61	syl5ibrcom 246	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑢 ∈ 𝐴) → (𝑧 = (𝐹‘𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
63	62	rexlimdva 3154	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (∃𝑢 ∈ 𝐴 𝑧 = (𝐹‘𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
64	63	imp 407	. . . . . . . . . . . . 13 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ ∃𝑢 ∈ 𝐴 𝑧 = (𝐹‘𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
65	33, 64	syldan 591	. . . . . . . . . . . 12 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑧 ∈ 𝐵) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
66	65	ralrimiva 3145	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))
67		dffo3 7088	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto→𝐵 ↔ ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)))
68	30, 66, 67	sylanbrc 583	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto→𝐵)
69		fodomfi 9308	. . . . . . . . . 10 ⊢ (((𝐴 ∖ {𝑦}) ∈ Fin ∧ (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto→𝐵) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
70	27, 68, 69	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
71	70	anassrs 468	. . . . . . . 8 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
72	71	expr 457	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝑥 ≠ 𝑦 → 𝐵 ≼ (𝐴 ∖ {𝑦})))
73	72	necon1bd 2957	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (¬ 𝐵 ≼ (𝐴 ∖ {𝑦}) → 𝑥 = 𝑦))
74	23, 73	mpd 15	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 = 𝑦)
75	74	ex 413	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
76	75	ralrimivva 3199	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
77		dff13 7238	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
78	3, 76, 77	sylanbrc 583	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴–1-1→𝐵)
79		df-f1o 6539	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
80	78, 1, 79	sylanbrc 583	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∃wrex 3069   ∖ cdif 3941   ⊆ wss 3944   ⊊ wpss 3945  {csn 4622   class class class wbr 5141   ↾ cres 5671  ⟶wf 6528  –1-1→wf1 6529  –onto→wfo 6530  –1-1-onto→wf1o 6531  ‘cfv 6532   ≈ cen 8919   ≼ cdom 8920   ≺ csdm 8921  Fincfn 8922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-om 7839  df-1o 8448  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926

This theorem is referenced by:  rneqdmfinf1o  9311  phpreu  36276