Theorem unielxp 7973

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 7970	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
2		elvvuni 5701	. . . 4 ⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)
3	2	adantr 480	. . 3 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → ∪ 𝐴 ∈ 𝐴)
4		simprl 771	. . . . . 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 6834	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴))
8	7	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ∈ 𝐵 ↔ (1st ‘𝐴) ∈ 𝐵))
9		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴))
10	9	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ∈ 𝐶 ↔ (2nd ‘𝐴) ∈ 𝐶))
11	8, 10	anbi12d 633	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
12	6, 11	anbi12d 633	. . . . . . . 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 633	. . . . . . 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 3540	. . . . . 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 4865	. . . . 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 234	. . . 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 7972	. . . . . 6 ⊢ (𝐵 × 𝐶) = {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶)}
19		df-rab 3391	. . . . . 6 ⊢ {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶)} = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))}
20	18, 19	eqtri 2760	. . . . 5 ⊢ (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))}
21	20	unieqi 4863	. . . 4 ⊢ ∪ (𝐵 × 𝐶) = ∪ {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) ∈ 𝐶))}
22	17, 21	eleqtrrdi 2848	. . 3 ⊢ ((∪ 𝐴 ∈ 𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))
23	3, 22	mpancom 689	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))
24	1, 23	sylbi 217	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  {crab 3390  Vcvv 3430  ∪ cuni 4851   × cxp 5622  ‘cfv 6492  1^st c1st 7933  2^nd c2nd 7934

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-1st 7935  df-2nd 7936

This theorem is referenced by: (None)