Theorem disjxp1 6215

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 6144	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 × 𝐶) → (1st ‘𝑦) ∈ 𝐵)
2		xp1st 6144	. . . . . . 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 3967	. . . . . . . . . . . 12 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
5	3, 4	sylib 121	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
6		1stexg 6146	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (1st ‘𝑦) ∈ V)
7	6	elv 2734	. . . . . . . . . . . 12 ⊢ (1st ‘𝑦) ∈ V
8		eleq1 2233	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘𝑦) → (𝑧 ∈ 𝐵 ↔ (1st ‘𝑦) ∈ 𝐵))
9	8	rmobidv 2658	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘𝑦) → (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑥 ∈ 𝐴 (1st ‘𝑦) ∈ 𝐵))
10	7, 9	spcv 2824	. . . . . . . . . . 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 2312	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐴
13		nfcv 2312	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝐴
14		nfcsb1v 3082	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
15	14	nfel2 2325	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵
16		csbeq1a 3058	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
17	16	eleq2d 2240	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((1st ‘𝑦) ∈ 𝐵 ↔ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵))
18	12, 13, 15, 17	rmo4f 2928	. . . . . . . . . 10 ⊢ (∃*𝑥 ∈ 𝐴 (1st ‘𝑦) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((1st ‘𝑦) ∈ 𝐵 ∧ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵) → 𝑥 = 𝑤))
19	11, 18	sylib 121	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((1st ‘𝑦) ∈ 𝐵 ∧ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵) → 𝑥 = 𝑤))
20	19	r19.21bi 2558	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑤 ∈ 𝐴 (((1st ‘𝑦) ∈ 𝐵 ∧ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵) → 𝑥 = 𝑤))
21	20	r19.21bi 2558	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) → (((1st ‘𝑦) ∈ 𝐵 ∧ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵) → 𝑥 = 𝑤))
22	1, 2, 21	syl2ani 406	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) → ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)) → 𝑥 = 𝑤))
23	22	ralrimiva 2543	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑤 ∈ 𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)) → 𝑥 = 𝑤))
24	23	ralrimiva 2543	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)) → 𝑥 = 𝑤))
25		nfcsb1v 3082	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐶
26	14, 25	nfxp 4638	. . . . . 6 ⊢ Ⅎ𝑥(⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)
27	26	nfel2 2325	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)
28		csbeq1a 3058	. . . . . . 7 ⊢ (𝑥 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑥⦌𝐶)
29	16, 28	xpeq12d 4636	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝐵 × 𝐶) = (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶))
30	29	eleq2d 2240	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑦 ∈ (𝐵 × 𝐶) ↔ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)))
31	12, 13, 27, 30	rmo4f 2928	. . . 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 133	. . 3 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 × 𝐶))
33	32	alrimiv 1867	. 2 ⊢ (𝜑 → ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 × 𝐶))
34		df-disj 3967	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 × 𝐶))
35	33, 34	sylibr 133	1 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 (𝐵 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1346   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ∃*wrmo 2451  Vcvv 2730  ⦋csb 3049  Disj wdisj 3966   × cxp 4609  ‘cfv 5198  1^st c1st 6117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rmo 2456  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-disj 3967  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-1st 6119

This theorem is referenced by:  disjsnxp  6216