Theorem disjf1 41463

Description: A 1 to 1 mapping built from disjoint, nonempty sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
disjf1.xph	⊢ Ⅎ𝑥𝜑
disjf1.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
disjf1.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
disjf1.n0	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
disjf1.dj	⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)

Assertion

Ref	Expression
disjf1	⊢ (𝜑 → 𝐹:𝐴–1-1→𝑉)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

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

Proof of Theorem disjf1

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

Step	Hyp	Ref	Expression
1		disjf1.xph	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
2		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	1, 2	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝐴)
4		nfcsb1v 3907	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑥𝑉
6	4, 5	nfel 2992	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉
7	3, 6	nfim 1897	. . . . 5 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉)
8		eleq1w 2895	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
9	8	anbi2d 630	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)))
10		csbeq1a 3897	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11	10	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ 𝑉 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉))
12	9, 11	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉)))
13		disjf1.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
14	7, 12, 13	chvarfv 2242	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉)
15	14	ralrimiva 3182	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉)
16		inidm 4195	. . . . . . . . 9 ⊢ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐵) = ⦋𝑦 / 𝑥⦌𝐵
17	16	eqcomi 2830	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑥⦌𝐵 = (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐵)
18	17	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → ⦋𝑦 / 𝑥⦌𝐵 = (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐵))
19		ineq2 4183	. . . . . . . 8 ⊢ (⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐵) = (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵))
20	19	ad2antlr 725	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐵) = (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵))
21		disjf1.dj	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
22		nfcv 2977	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝐵
23		nfcsb1v 3907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
24		csbeq1a 3897	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
25	22, 23, 24	cbvdisj 5041	. . . . . . . . . 10 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑤 ∈ 𝐴 ⦋𝑤 / 𝑥⦌𝐵)
26	21, 25	sylib 220	. . . . . . . . 9 ⊢ (𝜑 → Disj 𝑤 ∈ 𝐴 ⦋𝑤 / 𝑥⦌𝐵)
27	26	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → Disj 𝑤 ∈ 𝐴 ⦋𝑤 / 𝑥⦌𝐵)
28		simpllr 774	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
29		neqne 3024	. . . . . . . . 9 ⊢ (¬ 𝑦 = 𝑧 → 𝑦 ≠ 𝑧)
30	29	adantl 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 ≠ 𝑧)
31		csbeq1 3886	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ⦋𝑤 / 𝑥⦌𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
32		csbeq1 3886	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
33	31, 32	disji2 5048	. . . . . . . 8 ⊢ ((Disj 𝑤 ∈ 𝐴 ⦋𝑤 / 𝑥⦌𝐵 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 ≠ 𝑧) → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅)
34	27, 28, 30, 33	syl3anc 1367	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅)
35	18, 20, 34	3eqtrd 2860	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → ⦋𝑦 / 𝑥⦌𝐵 = ∅)
36		nfcv 2977	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∅
37	4, 36	nfne 3119	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ≠ ∅
38	3, 37	nfim 1897	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ≠ ∅)
39	10	neeq1d 3075	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐵 ≠ ∅ ↔ ⦋𝑦 / 𝑥⦌𝐵 ≠ ∅))
40	9, 39	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ≠ ∅)))
41		disjf1.n0	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
42	38, 40, 41	chvarfv 2242	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ≠ ∅)
43	42	adantrr 715	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ⦋𝑦 / 𝑥⦌𝐵 ≠ ∅)
44	43	ad2antrr 724	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → ⦋𝑦 / 𝑥⦌𝐵 ≠ ∅)
45	44	neneqd 3021	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → ¬ ⦋𝑦 / 𝑥⦌𝐵 = ∅)
46	35, 45	condan 816	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) → 𝑦 = 𝑧)
47	46	ex 415	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵 → 𝑦 = 𝑧))
48	47	ralrimivva 3191	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵 → 𝑦 = 𝑧))
49	15, 48	jca 514	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵 → 𝑦 = 𝑧)))
50		disjf1.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
51		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑦𝐵
52	51, 4, 10	cbvmpt 5167	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
53	50, 52	eqtri 2844	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
54		csbeq1 3886	. . 3 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
55	53, 54	f1mpt 7019	. 2 ⊢ (𝐹:𝐴–1-1→𝑉 ↔ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵 → 𝑦 = 𝑧)))
56	49, 55	sylibr 236	1 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ⦋csb 3883   ∩ cin 3935  ∅c0 4291  Disj wdisj 5031   ↦ cmpt 5146  –1-1→wf1 6352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-disj 5032  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fv 6363

This theorem is referenced by:  disjf1o  41472  meadjiunlem  42767