Theorem unielxp 5742

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 5739	. 2 ⊢ (A ∈ (B × 𝐶) ↔ (A ∈ (V × V) ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶)))
2		elvvuni 4347	. . . 4 ⊢ (A ∈ (V × V) → ∪ A ∈ A)
3	2	adantr 261	. . 3 ⊢ ((A ∈ (V × V) ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶)) → ∪ A ∈ A)
4		simprl 483	. . . . . 6 ⊢ ((∪ A ∈ A ∧ (A ∈ (V × V) ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶))) → A ∈ (V × V))
5		eleq2 2098	. . . . . . . 8 ⊢ (x = A → (∪ A ∈ x ↔ ∪ A ∈ A))
6		eleq1 2097	. . . . . . . . 9 ⊢ (x = A → (x ∈ (V × V) ↔ A ∈ (V × V)))
7		fveq2 5121	. . . . . . . . . . 11 ⊢ (x = A → (1st ‘x) = (1st ‘A))
8	7	eleq1d 2103	. . . . . . . . . 10 ⊢ (x = A → ((1st ‘x) ∈ B ↔ (1st ‘A) ∈ B))
9		fveq2 5121	. . . . . . . . . . 11 ⊢ (x = A → (2nd ‘x) = (2nd ‘A))
10	9	eleq1d 2103	. . . . . . . . . 10 ⊢ (x = A → ((2nd ‘x) ∈ 𝐶 ↔ (2nd ‘A) ∈ 𝐶))
11	8, 10	anbi12d 442	. . . . . . . . 9 ⊢ (x = A → (((1st ‘x) ∈ B ∧ (2nd ‘x) ∈ 𝐶) ↔ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶)))
12	6, 11	anbi12d 442	. . . . . . . 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 442	. . . . . . 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 2635	. . . . . 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 3583	. . . . 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 5741	. . . . . 6 ⊢ (B × 𝐶) = {x ∈ (V × V) ∣ ((1st ‘x) ∈ B ∧ (2nd ‘x) ∈ 𝐶)}
19		df-rab 2309	. . . . . 6 ⊢ {x ∈ (V × V) ∣ ((1st ‘x) ∈ B ∧ (2nd ‘x) ∈ 𝐶)} = {x ∣ (x ∈ (V × V) ∧ ((1st ‘x) ∈ B ∧ (2nd ‘x) ∈ 𝐶))}
20	18, 19	eqtri 2057	. . . . 5 ⊢ (B × 𝐶) = {x ∣ (x ∈ (V × V) ∧ ((1st ‘x) ∈ B ∧ (2nd ‘x) ∈ 𝐶))}
21	20	unieqi 3581	. . . 4 ⊢ ∪ (B × 𝐶) = ∪ {x ∣ (x ∈ (V × V) ∧ ((1st ‘x) ∈ B ∧ (2nd ‘x) ∈ 𝐶))}
22	17, 21	syl6eleqr 2128	. . 3 ⊢ ((∪ A ∈ A ∧ (A ∈ (V × V) ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶))) → ∪ A ∈ ∪ (B × 𝐶))
23	3, 22	mpancom 399	. 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 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  {crab 2304  Vcvv 2551  ∪ cuni 3571   × cxp 4286  ‘cfv 4845  1^st c1st 5707  2^nd c2nd 5708

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710

This theorem is referenced by: (None)