Theorem disjf1o 46314

Description: A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
disjf1o.xph	⊢ Ⅎ𝑥𝜑
disjf1o.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
disjf1o.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
disjf1o.dj	⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
disjf1o.d	⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
disjf1o.e	⊢ 𝐷 = (ran 𝐹 ∖ {∅})

Assertion

Ref	Expression
disjf1o	⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶–1-1-onto→𝐷)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑉

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem disjf1o

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjf1o.xph	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eqid 2760	. . . 4 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵)
3		simpl 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜑)
4		disjf1o.d	. . . . . . . 8 ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
5		ssrab2 4004	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ⊆ 𝐴
6	4, 5	eqsstri 3955	. . . . . . 7 ⊢ 𝐶 ⊆ 𝐴
7		id 23	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐶)
8	6, 7	sselid 3907	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴)
9	8	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐴)
10		disjf1o.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
11	3, 9, 10	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉)
12	7, 4	eleqtrdi 2870	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅})
13		rabid 3432	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
14	13	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ≠ ∅)))
15	12, 14	mpbid 235	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
16	15	simprd 501	. . . . 5 ⊢ (𝑥 ∈ 𝐶 → 𝐵 ≠ ∅)
17	16	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ≠ ∅)
18	6	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
19		disjf1o.dj	. . . . 5 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
20		disjss1 5075	. . . . 5 ⊢ (𝐶 ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ 𝐶 𝐵))
21	18, 19, 20	sylc 66	. . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐶 𝐵)
22	1, 2, 11, 17, 21	disjf1 46306	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐵):𝐶–1-1→𝑉)
23		f1f1orn 6841	. . 3 ⊢ ((𝑥 ∈ 𝐶 ↦ 𝐵):𝐶–1-1→𝑉 → (𝑥 ∈ 𝐶 ↦ 𝐵):𝐶–1-1-onto→ran (𝑥 ∈ 𝐶 ↦ 𝐵))
24	22, 23	syl 18	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐵):𝐶–1-1-onto→ran (𝑥 ∈ 𝐶 ↦ 𝐵))
25		disjf1o.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
26	25	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
27	26	reseq1d 5970	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶))
28	18	resmptd 6034	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ 𝐵))
29	27, 28	eqtrd 2795	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ 𝐵))
30		eqidd 2761	. . 3 ⊢ (𝜑 → 𝐶 = 𝐶)
31		simpl 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝜑)
32		id 23	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐷)
33		disjf1o.e	. . . . . . . . . 10 ⊢ 𝐷 = (ran 𝐹 ∖ {∅})
34	32, 33	eleqtrdi 2870	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ (ran 𝐹 ∖ {∅}))
35		eldifsni 4749	. . . . . . . . 9 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {∅}) → 𝑦 ≠ ∅)
36	34, 35	syl 18	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ≠ ∅)
37	36	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑦 ≠ ∅)
38		eldifi 4066	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {∅}) → 𝑦 ∈ ran 𝐹)
39	34, 38	syl 18	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ ran 𝐹)
40	25	elrnmpt 5941	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
41	39, 40	syl 18	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐷 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
42	39, 41	mpbid 235	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
43	42	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
44		nfv 1946	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ≠ ∅
45	1, 44	nfan 1931	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ≠ ∅)
46		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑦
47		nfmpt1 5203	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶 ↦ 𝐵)
48	47	nfrn 5935	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐶 ↦ 𝐵)
49	46, 48	nfel 2936	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵)
50		simp3 1156	. . . . . . . . . . . 12 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
51		simp2 1155	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐴)
52		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
53	52	eqcomd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝐵 → 𝐵 = 𝑦)
54	53	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝐵 = 𝑦)
55		simpl 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
56	54, 55	eqnetrd 3022	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
57	56	3adant2 1149	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
58	51, 57	jca 521	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
59	58, 13	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅})
60	4	eqcomi 2769	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅} = 𝐶
61	60	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅} = 𝐶)
62	59, 61	eleqtrd 2862	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐶)
63		eqvisset 3470	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V)
64	63	3ad2ant3 1153	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝐵 ∈ V)
65	2	elrnmpt1 5943	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝐵 ∈ V) → 𝐵 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
66	62, 64, 65	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝐵 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
67	50, 66	eqeltrd 2860	. . . . . . . . . . 11 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
68	67	3adant1l 1195	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ≠ ∅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
69	68	3exp 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ≠ ∅) → (𝑥 ∈ 𝐴 → (𝑦 = 𝐵 → 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))))
70	45, 49, 69	rexlimd 3269	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ≠ ∅) → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 → 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵)))
71	70	imp 412	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ≠ ∅) ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
72	31, 37, 43, 71	syl21anc 851	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
73	72	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐷 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
74		dfss3 3898	. . . . 5 ⊢ (𝐷 ⊆ ran (𝑥 ∈ 𝐶 ↦ 𝐵) ↔ ∀𝑦 ∈ 𝐷 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
75	73, 74	sylibr 237	. . . 4 ⊢ (𝜑 → 𝐷 ⊆ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
76		simpl 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵)) → 𝜑)
77		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
78	2	elrnmpt 5941	. . . . . . 7 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵))
79	77, 78	ax-mp 5	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵)
80	79	bilani 510	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵)) → ∃𝑥 ∈ 𝐶 𝑦 = 𝐵)
81		nfv 1946	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐷
82		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
83	8	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐴)
84	82, 63	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝐵 ∈ V)
85	25	elrnmpt1 5943	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → 𝐵 ∈ ran 𝐹)
86	83, 84, 85	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝐵 ∈ ran 𝐹)
87	82, 86	eqeltrd 2860	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 ∈ ran 𝐹)
88	87	3adant1 1148	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 ∈ ran 𝐹)
89	16	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
90	82, 89	eqnetrd 3022	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
91		nelsn 4623	. . . . . . . . . . . 12 ⊢ (𝑦 ≠ ∅ → ¬ 𝑦 ∈ {∅})
92	90, 91	syl 18	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → ¬ 𝑦 ∈ {∅})
93	92	3adant1 1148	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → ¬ 𝑦 ∈ {∅})
94	88, 93	eldifd 4049	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 ∈ (ran 𝐹 ∖ {∅}))
95	94, 33	eleqtrrdi 2871	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 ∈ 𝐷)
96	95	3exp 1137	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐶 → (𝑦 = 𝐵 → 𝑦 ∈ 𝐷)))
97	1, 81, 96	rexlimd 3269	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ 𝐶 𝑦 = 𝐵 → 𝑦 ∈ 𝐷))
98	97	imp 412	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵) → 𝑦 ∈ 𝐷)
99	76, 80, 98	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵)) → 𝑦 ∈ 𝐷)
100	75, 99	eqelssd 3930	. . 3 ⊢ (𝜑 → 𝐷 = ran (𝑥 ∈ 𝐶 ↦ 𝐵))
101	29, 30, 100	f1oeq123d 6823	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐶):𝐶–1-1-onto→𝐷 ↔ (𝑥 ∈ 𝐶 ↦ 𝐵):𝐶–1-1-onto→ran (𝑥 ∈ 𝐶 ↦ 𝐵)))
102	24, 101	mpbird 260	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶–1-1-onto→𝐷)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3896   ∖ cdif 4044  ∅c0 4277  {csn 4584  Disj wdisj 5069   ↦ cmpt 5185  ran crn 5652   ↾ cres 5653  –1-1→wf1 6540  –1-1-onto→wf1o 6542

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-disj 5070  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551

This theorem is used by:  sge0fodjrnlem  47534