Theorem fofinf1o 8875

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 1137	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴–onto→𝐵)
2		fof 6593	. . . 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 8696	. . . . . . 7 ⊢ (𝐵 ≼ (𝐴 ∖ {𝑦}) → ¬ (𝐴 ∖ {𝑦}) ≺ 𝐵)
5		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐵 ∈ Fin)
6		simp2 1138	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ≈ 𝐵)
7		enfii 8787	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
8	5, 6, 7	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin)
9	8	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐴 ∈ Fin)
10		difssd 4024	. . . . . . . . . 10 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ⊆ 𝐴)
11		simplrr 778	. . . . . . . . . . . 12 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ 𝐴)
12		neldifsn 4681	. . . . . . . . . . . 12 ⊢ ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})
13		nelne1 3031	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
14	11, 12, 13	sylancl 589	. . . . . . . . . . 11 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐴 ≠ (𝐴 ∖ {𝑦}))
15	14	necomd 2990	. . . . . . . . . 10 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
16		df-pss 3863	. . . . . . . . . 10 ⊢ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ ((𝐴 ∖ {𝑦}) ⊆ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≠ 𝐴))
17	10, 15, 16	sylanbrc 586	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
18		php3 8756	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊊ 𝐴) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
19	9, 17, 18	syl2anc 587	. . . . . . . 8 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐴)
20	6	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐴 ≈ 𝐵)
21		sdomentr 8704	. . . . . . . 8 ⊢ (((𝐴 ∖ {𝑦}) ≺ 𝐴 ∧ 𝐴 ≈ 𝐵) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
22	19, 20, 21	syl2anc 587	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐵)
23	4, 22	nsyl3 140	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → ¬ 𝐵 ≼ (𝐴 ∖ {𝑦}))
24	8	adantr 484	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝐴 ∈ Fin)
25		difss 4023	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝑦}) ⊆ 𝐴
26		ssfi 8775	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐴 ∖ {𝑦}) ∈ Fin)
27	24, 25, 26	sylancl 589	. . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝐹:𝐴⟶𝐵)
29		fssres 6545	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵)
30	28, 25, 29	sylancl 589	. . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑧 ∈ 𝐵) → ∃𝑢 ∈ 𝐴 𝑧 = (𝐹‘𝑢))
33	31, 32	sylan 583	. . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝑥 ∈ 𝐴)
35		simprrr 782	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝑥 ≠ 𝑦)
36		eldifsn 4676	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝑦))
37	34, 35, 36	sylanbrc 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝑥 ∈ (𝐴 ∖ {𝑦}))
38		simprrl 781	. . . . . . . . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . . . . . . . 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 6677	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
41	40	rspceeqv 3542	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹‘𝑦) = (𝐹‘𝑥)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑦) = (𝐹‘𝑤))
42	37, 39, 41	syl2anc 587	. . . . . . . . . . . . . . . . . . . 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 6686	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑦 → ((𝐹‘𝑢) = (𝐹‘𝑤) ↔ (𝐹‘𝑦) = (𝐹‘𝑤)))
44	43	rexbidv 3208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑦 → (∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑦) = (𝐹‘𝑤)))
45	42, 44	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . 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 4676	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑢 ∈ 𝐴 ∧ 𝑢 ≠ 𝑦))
49		eqid 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹‘𝑢) = (𝐹‘𝑢)
50		fveq2 6677	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑢 → (𝐹‘𝑤) = (𝐹‘𝑢))
51	50	rspceeqv 3542	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑢 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹‘𝑢) = (𝐹‘𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))
52	49, 51	mpan2 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ (𝐴 ∖ {𝑦}) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))
53	48, 52	sylbir 238	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑢 ≠ 𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))
54	53	adantll 714	. . . . . . . . . . . . . . . . 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 3022	. . . . . . . . . . . . . . . 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 6696	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (𝐴 ∖ {𝑦}) → ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) = (𝐹‘𝑤))
57	56	eqeq2d 2750	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝐴 ∖ {𝑦}) → (𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ 𝑧 = (𝐹‘𝑤)))
58	57	rexbiia 3161	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹‘𝑤))
59		eqeq1 2743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐹‘𝑢) → (𝑧 = (𝐹‘𝑤) ↔ (𝐹‘𝑢) = (𝐹‘𝑤)))
60	59	rexbidv 3208	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐹‘𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤)))
61	58, 60	syl5bb 286	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐹‘𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤)))
62	55, 61	syl5ibrcom 250	. . . . . . . . . . . . . . 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 3195	. . . . . . . . . . . . . 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 410	. . . . . . . . . . . . 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 594	. . . . . . . . . . . 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 3097	. . . . . . . . . . 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 6881	. . . . . . . . . . 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 586	. . . . . . . . . 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 8873	. . . . . . . . . 10 ⊢ (((𝐴 ∖ {𝑦}) ∈ Fin ∧ (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto→𝐵) → 𝐵 ≼ (𝐴 ∖ {𝑦}))
70	27, 68, 69	syl2anc 587	. . . . . . . . 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 471	. . . . . . . 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 460	. . . . . . 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 2953	. . . . . 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 416	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
76	75	ralrimivva 3104	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
77		dff13 7027	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
78	3, 76, 77	sylanbrc 586	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴–1-1→𝐵)
79		df-f1o 6347	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
80	78, 1, 79	sylanbrc 586	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114   ≠ wne 2935  ∀wral 3054  ∃wrex 3055   ∖ cdif 3841   ⊆ wss 3844   ⊊ wpss 3845  {csn 4517   class class class wbr 5031   ↾ cres 5528  ⟶wf 6336  –1-1→wf1 6337  –onto→wfo 6338  –1-1-onto→wf1o 6339  ‘cfv 6340   ≈ cen 8555   ≼ cdom 8556   ≺ csdm 8557  Fincfn 8558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-om 7603  df-1o 8134  df-er 8323  df-en 8559  df-dom 8560  df-sdom 8561  df-fin 8562

This theorem is referenced by:  rneqdmfinf1o  8876  phpreu  35407