Theorem disjxp1 6204

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 6133	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 × 𝐶) → (1st ‘𝑦) ∈ 𝐵)
2		xp1st 6133	. . . . . . 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 3960	. . . . . . . . . . . 12 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
5	3, 4	sylib 121	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
6		1stexg 6135	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (1st ‘𝑦) ∈ V)
7	6	elv 2730	. . . . . . . . . . . 12 ⊢ (1st ‘𝑦) ∈ V
8		eleq1 2229	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘𝑦) → (𝑧 ∈ 𝐵 ↔ (1st ‘𝑦) ∈ 𝐵))
9	8	rmobidv 2654	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘𝑦) → (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑥 ∈ 𝐴 (1st ‘𝑦) ∈ 𝐵))
10	7, 9	spcv 2820	. . . . . . . . . . 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 2308	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐴
13		nfcv 2308	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝐴
14		nfcsb1v 3078	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
15	14	nfel2 2321	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵
16		csbeq1a 3054	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
17	16	eleq2d 2236	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((1st ‘𝑦) ∈ 𝐵 ↔ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵))
18	12, 13, 15, 17	rmo4f 2924	. . . . . . . . . 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 2554	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑤 ∈ 𝐴 (((1st ‘𝑦) ∈ 𝐵 ∧ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵) → 𝑥 = 𝑤))
21	20	r19.21bi 2554	. . . . . . 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 2539	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑤 ∈ 𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)) → 𝑥 = 𝑤))
24	23	ralrimiva 2539	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)) → 𝑥 = 𝑤))
25		nfcsb1v 3078	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐶
26	14, 25	nfxp 4631	. . . . . 6 ⊢ Ⅎ𝑥(⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)
27	26	nfel2 2321	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)
28		csbeq1a 3054	. . . . . . 7 ⊢ (𝑥 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑥⦌𝐶)
29	16, 28	xpeq12d 4629	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝐵 × 𝐶) = (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶))
30	29	eleq2d 2236	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑦 ∈ (𝐵 × 𝐶) ↔ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)))
31	12, 13, 27, 30	rmo4f 2924	. . . 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 1862	. 2 ⊢ (𝜑 → ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 × 𝐶))
34		df-disj 3960	. 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 1341   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ∃*wrmo 2447  Vcvv 2726  ⦋csb 3045  Disj wdisj 3959   × cxp 4602  ‘cfv 5188  1^st c1st 6106

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rmo 2452  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-disj 3960  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-1st 6108

This theorem is referenced by:  disjsnxp  6205