Theorem disjf1 45090

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 1913	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	1, 2	nfan 1898	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝐴)
4		nfcsb1v 3946	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥𝑉
6	4, 5	nfel 2923	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉
7	3, 6	nfim 1895	. . . . 5 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉)
8		eleq1w 2827	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
9	8	anbi2d 629	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)))
10		csbeq1a 3935	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11	10	eleq1d 2829	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ 𝑉 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉))
12	9, 11	imbi12d 344	. . . . 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 2241	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉)
15	14	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉)
16		inidm 4248	. . . . . . . . 9 ⊢ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐵) = ⦋𝑦 / 𝑥⦌𝐵
17	16	eqcomi 2749	. . . . . . . 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 4235	. . . . . . . 8 ⊢ (⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐵) = (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵))
20	19	ad2antlr 726	. . . . . . 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 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝐵
23		nfcsb1v 3946	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
24		csbeq1a 3935	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
25	22, 23, 24	cbvdisj 5143	. . . . . . . . . 10 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑤 ∈ 𝐴 ⦋𝑤 / 𝑥⦌𝐵)
26	21, 25	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → Disj 𝑤 ∈ 𝐴 ⦋𝑤 / 𝑥⦌𝐵)
27	26	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → Disj 𝑤 ∈ 𝐴 ⦋𝑤 / 𝑥⦌𝐵)
28		simpllr 775	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
29		neqne 2954	. . . . . . . . 9 ⊢ (¬ 𝑦 = 𝑧 → 𝑦 ≠ 𝑧)
30	29	adantl 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 ≠ 𝑧)
31		csbeq1 3924	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ⦋𝑤 / 𝑥⦌𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
32		csbeq1 3924	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
33	31, 32	disji2 5150	. . . . . . . 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 1371	. . . . . . 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 2784	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → ⦋𝑦 / 𝑥⦌𝐵 = ∅)
36		nfcv 2908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∅
37	4, 36	nfne 3049	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ≠ ∅
38	3, 37	nfim 1895	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ≠ ∅)
39	10	neeq1d 3006	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐵 ≠ ∅ ↔ ⦋𝑦 / 𝑥⦌𝐵 ≠ ∅))
40	9, 39	imbi12d 344	. . . . . . . . . 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 2241	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ≠ ∅)
43	42	adantrr 716	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ⦋𝑦 / 𝑥⦌𝐵 ≠ ∅)
44	43	ad2antrr 725	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → ⦋𝑦 / 𝑥⦌𝐵 ≠ ∅)
45	44	neneqd 2951	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ∧ ¬ 𝑦 = 𝑧) → ¬ ⦋𝑦 / 𝑥⦌𝐵 = ∅)
46	35, 45	condan 817	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) → 𝑦 = 𝑧)
47	46	ex 412	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵 → 𝑦 = 𝑧))
48	47	ralrimivva 3208	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵 → 𝑦 = 𝑧))
49	15, 48	jca 511	. 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 2908	. . . . 5 ⊢ Ⅎ𝑦𝐵
52	51, 4, 10	cbvmpt 5277	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
53	50, 52	eqtri 2768	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
54		csbeq1 3924	. . 3 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
55	53, 54	f1mpt 7298	. 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 234	1 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ⦋csb 3921   ∩ cin 3975  ∅c0 4352  Disj wdisj 5133   ↦ cmpt 5249  –1-1→wf1 6570

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-fv 6581

This theorem is referenced by:  disjf1o  45098  meadjiunlem  46386