Theorem unielxp 6065

Description: The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)

Assertion

Ref	Expression
unielxp	⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))

Proof of Theorem unielxp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp7 6061	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
2		elvvuni 4598	. . . 4 ⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)
3	2	adantr 274	. . 3 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → ∪ 𝐴 ∈ 𝐴)
4		simprl 520	. . . . . 6 ⊢ ((∪ 𝐴 ∈ 𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) → 𝐴 ∈ (V × V))
5		eleq2 2201	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (∪ 𝐴 ∈ 𝑥 ↔ ∪ 𝐴 ∈ 𝐴))
6		eleq1 2200	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (V × V) ↔ 𝐴 ∈ (V × V)))
7		fveq2 5414	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴))
8	7	eleq1d 2206	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ∈ 𝐵 ↔ (1st ‘𝐴) ∈ 𝐵))
9		fveq2 5414	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴))
10	9	eleq1d 2206	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ∈ 𝐶 ↔ (2nd ‘𝐴) ∈ 𝐶))
11	8, 10	anbi12d 464	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
12	6, 11	anbi12d 464	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶)) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))))
13	5, 12	anbi12d 464	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((∪ 𝐴 ∈ 𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))) ↔ (∪ 𝐴 ∈ 𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))))
14	13	spcegv 2769	. . . . . 6 ⊢ (𝐴 ∈ (V × V) → ((∪ 𝐴 ∈ 𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) → ∃𝑥(∪ 𝐴 ∈ 𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶)))))
15	4, 14	mpcom 36	. . . . 5 ⊢ ((∪ 𝐴 ∈ 𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) → ∃𝑥(∪ 𝐴 ∈ 𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))))
16		eluniab 3743	. . . . 5 ⊢ (∪ 𝐴 ∈ ∪ {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))} ↔ ∃𝑥(∪ 𝐴 ∈ 𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))))
17	15, 16	sylibr 133	. . . 4 ⊢ ((∪ 𝐴 ∈ 𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) → ∪ 𝐴 ∈ ∪ {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))})
18		xp2 6064	. . . . . 6 ⊢ (𝐵 × 𝐶) = {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶)}
19		df-rab 2423	. . . . . 6 ⊢ {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶)} = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))}
20	18, 19	eqtri 2158	. . . . 5 ⊢ (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))}
21	20	unieqi 3741	. . . 4 ⊢ ∪ (𝐵 × 𝐶) = ∪ {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))}
22	17, 21	eleqtrrdi 2231	. . 3 ⊢ ((∪ 𝐴 ∈ 𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))
23	3, 22	mpancom 418	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))
24	1, 23	sylbi 120	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {cab 2123  {crab 2418  Vcvv 2681  ∪ cuni 3731   × cxp 4532  ‘cfv 5118  1^st c1st 6029  2^nd c2nd 6030

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-1st 6031  df-2nd 6032

This theorem is referenced by: (None)