Theorem unielxp 5723

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 5720	. 2 ⊢ (A ∈ (B × 𝐶) ↔ (A ∈ (V × V) ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶)))
2		elvvuni 4331	. . . 4 ⊢ (A ∈ (V × V) → ∪ A ∈ A)
3	2	adantr 261	. . 3 ⊢ ((A ∈ (V × V) ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶)) → ∪ A ∈ A)
4		simprl 471	. . . . . 6 ⊢ ((∪ A ∈ A ∧ (A ∈ (V × V) ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶))) → A ∈ (V × V))
5		eleq2 2083	. . . . . . . 8 ⊢ (x = A → (∪ A ∈ x ↔ ∪ A ∈ A))
6		eleq1 2082	. . . . . . . . 9 ⊢ (x = A → (x ∈ (V × V) ↔ A ∈ (V × V)))
7		fveq2 5103	. . . . . . . . . . 11 ⊢ (x = A → (1st ‘x) = (1st ‘A))
8	7	eleq1d 2088	. . . . . . . . . 10 ⊢ (x = A → ((1st ‘x) ∈ B ↔ (1st ‘A) ∈ B))
9		fveq2 5103	. . . . . . . . . . 11 ⊢ (x = A → (2nd ‘x) = (2nd ‘A))
10	9	eleq1d 2088	. . . . . . . . . 10 ⊢ (x = A → ((2nd ‘x) ∈ 𝐶 ↔ (2nd ‘A) ∈ 𝐶))
11	8, 10	anbi12d 445	. . . . . . . . 9 ⊢ (x = A → (((1st ‘x) ∈ B ∧ (2nd ‘x) ∈ 𝐶) ↔ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶)))
12	6, 11	anbi12d 445	. . . . . . . 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 445	. . . . . . 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 2618	. . . . . 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 32	. . . . 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 3566	. . . . 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 137	. . . 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 5722	. . . . . 6 ⊢ (B × 𝐶) = {x ∈ (V × V) ∣ ((1st ‘x) ∈ B ∧ (2nd ‘x) ∈ 𝐶)}
19		df-rab 2293	. . . . . 6 ⊢ {x ∈ (V × V) ∣ ((1st ‘x) ∈ B ∧ (2nd ‘x) ∈ 𝐶)} = {x ∣ (x ∈ (V × V) ∧ ((1st ‘x) ∈ B ∧ (2nd ‘x) ∈ 𝐶))}
20	18, 19	eqtri 2042	. . . . 5 ⊢ (B × 𝐶) = {x ∣ (x ∈ (V × V) ∧ ((1st ‘x) ∈ B ∧ (2nd ‘x) ∈ 𝐶))}
21	20	unieqi 3564	. . . 4 ⊢ ∪ (B × 𝐶) = ∪ {x ∣ (x ∈ (V × V) ∧ ((1st ‘x) ∈ B ∧ (2nd ‘x) ∈ 𝐶))}
22	17, 21	syl6eleqr 2113	. . 3 ⊢ ((∪ A ∈ A ∧ (A ∈ (V × V) ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶))) → ∪ A ∈ ∪ (B × 𝐶))
23	3, 22	mpancom 401	. 2 ⊢ ((A ∈ (V × V) ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶)) → ∪ A ∈ ∪ (B × 𝐶))
24	1, 23	sylbi 114	1 ⊢ (A ∈ (B × 𝐶) → ∪ A ∈ ∪ (B × 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228  ∃wex 1362   ∈ wcel 1374  {cab 2008  {crab 2288  Vcvv 2535  ∪ cuni 3554   × cxp 4270  ‘cfv 4829  1^st c1st 5688  2^nd c2nd 5689

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-1st 5690  df-2nd 5691

This theorem is referenced by: (None)