Theorem disjf1o 45098

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 2740	. . . 4 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵)
3		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜑)
4		disjf1o.d	. . . . . . . 8 ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
5		ssrab2 4103	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ⊆ 𝐴
6	4, 5	eqsstri 4043	. . . . . . 7 ⊢ 𝐶 ⊆ 𝐴
7		id 22	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐶)
8	6, 7	sselid 4006	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴)
9	8	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐴)
10		disjf1o.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
11	3, 9, 10	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉)
12	7, 4	eleqtrdi 2854	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅})
13		rabid 3465	. . . . . . . 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 232	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
16	15	simprd 495	. . . . 5 ⊢ (𝑥 ∈ 𝐶 → 𝐵 ≠ ∅)
17	16	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ≠ ∅)
18	6	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
19		disjf1o.dj	. . . . 5 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
20		disjss1 5139	. . . . 5 ⊢ (𝐶 ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ 𝐶 𝐵))
21	18, 19, 20	sylc 65	. . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐶 𝐵)
22	1, 2, 11, 17, 21	disjf1 45090	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐵):𝐶–1-1→𝑉)
23		f1f1orn 6873	. . 3 ⊢ ((𝑥 ∈ 𝐶 ↦ 𝐵):𝐶–1-1→𝑉 → (𝑥 ∈ 𝐶 ↦ 𝐵):𝐶–1-1-onto→ran (𝑥 ∈ 𝐶 ↦ 𝐵))
24	22, 23	syl 17	. 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 6008	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶))
28	18	resmptd 6069	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ 𝐵))
29	27, 28	eqtrd 2780	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ 𝐵))
30		eqidd 2741	. . 3 ⊢ (𝜑 → 𝐶 = 𝐶)
31		simpl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝜑)
32		id 22	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐷)
33		disjf1o.e	. . . . . . . . . 10 ⊢ 𝐷 = (ran 𝐹 ∖ {∅})
34	32, 33	eleqtrdi 2854	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ (ran 𝐹 ∖ {∅}))
35		eldifsni 4815	. . . . . . . . 9 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {∅}) → 𝑦 ≠ ∅)
36	34, 35	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ≠ ∅)
37	36	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑦 ≠ ∅)
38		eldifi 4154	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {∅}) → 𝑦 ∈ ran 𝐹)
39	34, 38	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ ran 𝐹)
40	25	elrnmpt 5981	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
41	39, 40	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐷 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
42	39, 41	mpbid 232	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
43	42	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
44		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ≠ ∅
45	1, 44	nfan 1898	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ≠ ∅)
46		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑦
47		nfmpt1 5274	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶 ↦ 𝐵)
48	47	nfrn 5977	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐶 ↦ 𝐵)
49	46, 48	nfel 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵)
50		simp3 1138	. . . . . . . . . . . 12 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
51		simp2 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐴)
52		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
53	52	eqcomd 2746	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝐵 → 𝐵 = 𝑦)
54	53	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝐵 = 𝑦)
55		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
56	54, 55	eqnetrd 3014	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
57	56	3adant2 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
58	51, 57	jca 511	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
59	58, 13	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅})
60	4	eqcomi 2749	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅} = 𝐶
61	60	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅} = 𝐶)
62	59, 61	eleqtrd 2846	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐶)
63		eqvisset 3508	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V)
64	63	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝐵 ∈ V)
65	2	elrnmpt1 5983	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝐵 ∈ V) → 𝐵 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
66	62, 64, 65	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝐵 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
67	50, 66	eqeltrd 2844	. . . . . . . . . . 11 ⊢ ((𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
68	67	3adant1l 1176	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ≠ ∅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
69	68	3exp 1119	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ≠ ∅) → (𝑥 ∈ 𝐴 → (𝑦 = 𝐵 → 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))))
70	45, 49, 69	rexlimd 3272	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ≠ ∅) → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 → 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵)))
71	70	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ≠ ∅) ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
72	31, 37, 43, 71	syl21anc 837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
73	72	ralrimiva 3152	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐷 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
74		dfss3 3997	. . . . 5 ⊢ (𝐷 ⊆ ran (𝑥 ∈ 𝐶 ↦ 𝐵) ↔ ∀𝑦 ∈ 𝐷 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
75	73, 74	sylibr 234	. . . 4 ⊢ (𝜑 → 𝐷 ⊆ ran (𝑥 ∈ 𝐶 ↦ 𝐵))
76		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵)) → 𝜑)
77		vex 3492	. . . . . . . 8 ⊢ 𝑦 ∈ V
78	2	elrnmpt 5981	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵))
79	77, 78	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵)
80	79	biimpi 216	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵) → ∃𝑥 ∈ 𝐶 𝑦 = 𝐵)
81	80	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵)) → ∃𝑥 ∈ 𝐶 𝑦 = 𝐵)
82		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐷
83		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
84	8	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐴)
85	83, 63	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝐵 ∈ V)
86	25	elrnmpt1 5983	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → 𝐵 ∈ ran 𝐹)
87	84, 85, 86	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝐵 ∈ ran 𝐹)
88	83, 87	eqeltrd 2844	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 ∈ ran 𝐹)
89	88	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 ∈ ran 𝐹)
90	16	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
91	83, 90	eqnetrd 3014	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
92		nelsn 4688	. . . . . . . . . . . 12 ⊢ (𝑦 ≠ ∅ → ¬ 𝑦 ∈ {∅})
93	91, 92	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → ¬ 𝑦 ∈ {∅})
94	93	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → ¬ 𝑦 ∈ {∅})
95	89, 94	eldifd 3987	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 ∈ (ran 𝐹 ∖ {∅}))
96	95, 33	eleqtrrdi 2855	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 ∈ 𝐷)
97	96	3exp 1119	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐶 → (𝑦 = 𝐵 → 𝑦 ∈ 𝐷)))
98	1, 82, 97	rexlimd 3272	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ 𝐶 𝑦 = 𝐵 → 𝑦 ∈ 𝐷))
99	98	imp 406	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵) → 𝑦 ∈ 𝐷)
100	76, 81, 99	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐶 ↦ 𝐵)) → 𝑦 ∈ 𝐷)
101	75, 100	eqelssd 4030	. . 3 ⊢ (𝜑 → 𝐷 = ran (𝑥 ∈ 𝐶 ↦ 𝐵))
102	29, 30, 101	f1oeq123d 6856	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐶):𝐶–1-1-onto→𝐷 ↔ (𝑥 ∈ 𝐶 ↦ 𝐵):𝐶–1-1-onto→ran (𝑥 ∈ 𝐶 ↦ 𝐵)))
103	24, 102	mpbird 257	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶–1-1-onto→𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352  {csn 4648  Disj wdisj 5133   ↦ cmpt 5249  ran crn 5701   ↾ cres 5702  –1-1→wf1 6570  –1-1-onto→wf1o 6572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by:  sge0fodjrnlem  46337