Theorem disjxp1 6410

Description: The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypothesis

Ref	Expression
disjxp1.1	⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)

Assertion

Ref	Expression
disjxp1	⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 (𝐵 × 𝐶))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem disjxp1

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

Step	Hyp	Ref	Expression
1		xp1st 6337	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 × 𝐶) → (1st ‘𝑦) ∈ 𝐵)
2		xp1st 6337	. . . . . . 7 ⊢ (𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶) → (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵)
3		disjxp1.1	. . . . . . . . . . . 12 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
4		df-disj 4070	. . . . . . . . . . . 12 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
5	3, 4	sylib 122	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
6		1stexg 6339	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (1st ‘𝑦) ∈ V)
7	6	elv 2807	. . . . . . . . . . . 12 ⊢ (1st ‘𝑦) ∈ V
8		eleq1 2294	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘𝑦) → (𝑧 ∈ 𝐵 ↔ (1st ‘𝑦) ∈ 𝐵))
9	8	rmobidv 2724	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘𝑦) → (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑥 ∈ 𝐴 (1st ‘𝑦) ∈ 𝐵))
10	7, 9	spcv 2901	. . . . . . . . . . 11 ⊢ (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 (1st ‘𝑦) ∈ 𝐵)
11	5, 10	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (1st ‘𝑦) ∈ 𝐵)
12		nfcv 2375	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐴
13		nfcv 2375	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝐴
14		nfcsb1v 3161	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
15	14	nfel2 2388	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵
16		csbeq1a 3137	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
17	16	eleq2d 2301	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((1st ‘𝑦) ∈ 𝐵 ↔ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵))
18	12, 13, 15, 17	rmo4f 3005	. . . . . . . . . 10 ⊢ (∃*𝑥 ∈ 𝐴 (1st ‘𝑦) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((1st ‘𝑦) ∈ 𝐵 ∧ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵) → 𝑥 = 𝑤))
19	11, 18	sylib 122	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((1st ‘𝑦) ∈ 𝐵 ∧ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵) → 𝑥 = 𝑤))
20	19	r19.21bi 2621	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑤 ∈ 𝐴 (((1st ‘𝑦) ∈ 𝐵 ∧ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵) → 𝑥 = 𝑤))
21	20	r19.21bi 2621	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) → (((1st ‘𝑦) ∈ 𝐵 ∧ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵) → 𝑥 = 𝑤))
22	1, 2, 21	syl2ani 408	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) → ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)) → 𝑥 = 𝑤))
23	22	ralrimiva 2606	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑤 ∈ 𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)) → 𝑥 = 𝑤))
24	23	ralrimiva 2606	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)) → 𝑥 = 𝑤))
25		nfcsb1v 3161	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐶
26	14, 25	nfxp 4758	. . . . . 6 ⊢ Ⅎ𝑥(⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)
27	26	nfel2 2388	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)
28		csbeq1a 3137	. . . . . . 7 ⊢ (𝑥 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑥⦌𝐶)
29	16, 28	xpeq12d 4756	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝐵 × 𝐶) = (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶))
30	29	eleq2d 2301	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑦 ∈ (𝐵 × 𝐶) ↔ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)))
31	12, 13, 27, 30	rmo4f 3005	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 × 𝐶) ↔ ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)) → 𝑥 = 𝑤))
32	24, 31	sylibr 134	. . 3 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 × 𝐶))
33	32	alrimiv 1922	. 2 ⊢ (𝜑 → ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 × 𝐶))
34		df-disj 4070	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 × 𝐶))
35	33, 34	sylibr 134	1 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 (𝐵 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1396   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃*wrmo 2514  Vcvv 2803  ⦋csb 3128  Disj wdisj 4069   × cxp 4729  ‘cfv 5333  1^st c1st 6310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rmo 2519  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-disj 4070  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-1st 6312

This theorem is referenced by:  disjsnxp  6411