Theorem unielxp 7095

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 7092	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
2		elvvuni 5102	. . . 4 ⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)
3	2	adantr 480	. . 3 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → ∪ 𝐴 ∈ 𝐴)
4		simprl 790	. . . . . 6 ⊢ ((∪ 𝐴 ∈ 𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) → 𝐴 ∈ (V × V))
5		eleq2 2677	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (∪ 𝐴 ∈ 𝑥 ↔ ∪ 𝐴 ∈ 𝐴))
6		eleq1 2676	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (V × V) ↔ 𝐴 ∈ (V × V)))
7		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴))
8	7	eleq1d 2672	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ∈ 𝐵 ↔ (1st ‘𝐴) ∈ 𝐵))
9		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴))
10	9	eleq1d 2672	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ∈ 𝐶 ↔ (2nd ‘𝐴) ∈ 𝐶))
11	8, 10	anbi12d 743	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
12	6, 11	anbi12d 743	. . . . . . . 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 743	. . . . . . 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 3267	. . . . . 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 37	. . . . 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 4383	. . . . 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 223	. . . 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 7094	. . . . . 6 ⊢ (𝐵 × 𝐶) = {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶)}
19		df-rab 2905	. . . . . 6 ⊢ {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶)} = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))}
20	18, 19	eqtri 2632	. . . . 5 ⊢ (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))}
21	20	unieqi 4381	. . . 4 ⊢ ∪ (𝐵 × 𝐶) = ∪ {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))}
22	17, 21	syl6eleqr 2699	. . 3 ⊢ ((∪ 𝐴 ∈ 𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))
23	3, 22	mpancom 700	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))
24	1, 23	sylbi 206	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173  ∪ cuni 4372   × cxp 5036  ‘cfv 5804  1^st c1st 7057  2^nd c2nd 7058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060

This theorem is referenced by: (None)