Theorem unielxp 7959

Description: The membership relation for a Cartesian 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 7956	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
2		elvvuni 5708	. . . 4 ⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)
3	2	adantr 481	. . 3 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → ∪ 𝐴 ∈ 𝐴)
4		simprl 769	. . . . . 6 ⊢ ((∪ 𝐴 ∈ 𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) → 𝐴 ∈ (V × V))
5		eleq2 2826	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (∪ 𝐴 ∈ 𝑥 ↔ ∪ 𝐴 ∈ 𝐴))
6		eleq1 2825	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (V × V) ↔ 𝐴 ∈ (V × V)))
7		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴))
8	7	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ∈ 𝐵 ↔ (1st ‘𝐴) ∈ 𝐵))
9		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴))
10	9	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ∈ 𝐶 ↔ (2nd ‘𝐴) ∈ 𝐶))
11	8, 10	anbi12d 631	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
12	6, 11	anbi12d 631	. . . . . . . 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 631	. . . . . . 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 3556	. . . . . 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 38	. . . . 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 4880	. . . . 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 233	. . . 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 7958	. . . . . 6 ⊢ (𝐵 × 𝐶) = {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶)}
19		df-rab 3408	. . . . . 6 ⊢ {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶)} = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))}
20	18, 19	eqtri 2764	. . . . 5 ⊢ (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))}
21	20	unieqi 4878	. . . 4 ⊢ ∪ (𝐵 × 𝐶) = ∪ {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))}
22	17, 21	eleqtrrdi 2849	. . 3 ⊢ ((∪ 𝐴 ∈ 𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))
23	3, 22	mpancom 686	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))
24	1, 23	sylbi 216	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2713  {crab 3407  Vcvv 3445  ∪ cuni 4865   × cxp 5631  ‘cfv 6496  1^st c1st 7919  2^nd c2nd 7920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fv 6504  df-1st 7921  df-2nd 7922

This theorem is referenced by: (None)