Theorem disjxp1 6141

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 6071	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 × 𝐶) → (1st ‘𝑦) ∈ 𝐵)
2		xp1st 6071	. . . . . . 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 3915	. . . . . . . . . . . 12 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
5	3, 4	sylib 121	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
6		1stexg 6073	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (1st ‘𝑦) ∈ V)
7	6	elv 2693	. . . . . . . . . . . 12 ⊢ (1st ‘𝑦) ∈ V
8		eleq1 2203	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘𝑦) → (𝑧 ∈ 𝐵 ↔ (1st ‘𝑦) ∈ 𝐵))
9	8	rmobidv 2622	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘𝑦) → (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑥 ∈ 𝐴 (1st ‘𝑦) ∈ 𝐵))
10	7, 9	spcv 2783	. . . . . . . . . . 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 2282	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐴
13		nfcv 2282	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝐴
14		nfcsb1v 3040	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
15	14	nfel2 2295	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵
16		csbeq1a 3016	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
17	16	eleq2d 2210	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((1st ‘𝑦) ∈ 𝐵 ↔ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵))
18	12, 13, 15, 17	rmo4f 2886	. . . . . . . . . 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 2523	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑤 ∈ 𝐴 (((1st ‘𝑦) ∈ 𝐵 ∧ (1st ‘𝑦) ∈ ⦋𝑤 / 𝑥⦌𝐵) → 𝑥 = 𝑤))
21	20	r19.21bi 2523	. . . . . . 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 2508	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑤 ∈ 𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)) → 𝑥 = 𝑤))
24	23	ralrimiva 2508	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ (𝐵 × 𝐶) ∧ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)) → 𝑥 = 𝑤))
25		nfcsb1v 3040	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐶
26	14, 25	nfxp 4574	. . . . . 6 ⊢ Ⅎ𝑥(⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)
27	26	nfel2 2295	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)
28		csbeq1a 3016	. . . . . . 7 ⊢ (𝑥 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑥⦌𝐶)
29	16, 28	xpeq12d 4572	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝐵 × 𝐶) = (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶))
30	29	eleq2d 2210	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑦 ∈ (𝐵 × 𝐶) ↔ 𝑦 ∈ (⦋𝑤 / 𝑥⦌𝐵 × ⦋𝑤 / 𝑥⦌𝐶)))
31	12, 13, 27, 30	rmo4f 2886	. . . 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 1847	. 2 ⊢ (𝜑 → ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 × 𝐶))
34		df-disj 3915	. 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 1330   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃*wrmo 2420  Vcvv 2689  ⦋csb 3007  Disj wdisj 3914   × cxp 4545  ‘cfv 5131  1^st c1st 6044

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rmo 2425  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-disj 3915  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137  df-fv 5139  df-1st 6046

This theorem is referenced by:  disjsnxp  6142