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Theorem unielxp 7248
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.
StepHypRef Expression
1 elxp7 7245 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
2 elvvuni 5213 . . . 4 (𝐴 ∈ (V × V) → 𝐴𝐴)
32adantr 480 . . 3 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴𝐴)
4 simprl 809 . . . . . 6 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 ∈ (V × V))
5 eleq2 2719 . . . . . . . 8 (𝑥 = 𝐴 → ( 𝐴𝑥 𝐴𝐴))
6 eleq1 2718 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 ∈ (V × V) ↔ 𝐴 ∈ (V × V)))
7 fveq2 6229 . . . . . . . . . . 11 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
87eleq1d 2715 . . . . . . . . . 10 (𝑥 = 𝐴 → ((1st𝑥) ∈ 𝐵 ↔ (1st𝐴) ∈ 𝐵))
9 fveq2 6229 . . . . . . . . . . 11 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
109eleq1d 2715 . . . . . . . . . 10 (𝑥 = 𝐴 → ((2nd𝑥) ∈ 𝐶 ↔ (2nd𝐴) ∈ 𝐶))
118, 10anbi12d 747 . . . . . . . . 9 (𝑥 = 𝐴 → (((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶) ↔ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
126, 11anbi12d 747 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))))
135, 12anbi12d 747 . . . . . . 7 (𝑥 = 𝐴 → (( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))) ↔ ( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))))
1413spcegv 3325 . . . . . 6 (𝐴 ∈ (V × V) → (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → ∃𝑥( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)))))
154, 14mpcom 38 . . . . 5 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → ∃𝑥( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))))
16 eluniab 4479 . . . . 5 ( 𝐴 {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))} ↔ ∃𝑥( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))))
1715, 16sylibr 224 . . . 4 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))})
18 xp2 7247 . . . . . 6 (𝐵 × 𝐶) = {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)}
19 df-rab 2950 . . . . . 6 {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)} = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2018, 19eqtri 2673 . . . . 5 (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2120unieqi 4477 . . . 4 (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2217, 21syl6eleqr 2741 . . 3 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 (𝐵 × 𝐶))
233, 22mpancom 704 . 2 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴 (𝐵 × 𝐶))
241, 23sylbi 207 1 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  {crab 2945  Vcvv 3231   cuni 4468   × cxp 5141  cfv 5926  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-1st 7210  df-2nd 7211
This theorem is referenced by: (None)
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