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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.
StepHypRef Expression
1 elxp7 7092 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
2 elvvuni 5102 . . . 4 (𝐴 ∈ (V × V) → 𝐴𝐴)
32adantr 480 . . 3 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴𝐴)
4 simprl 790 . . . . . 6 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 ∈ (V × V))
5 eleq2 2677 . . . . . . . 8 (𝑥 = 𝐴 → ( 𝐴𝑥 𝐴𝐴))
6 eleq1 2676 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 ∈ (V × V) ↔ 𝐴 ∈ (V × V)))
7 fveq2 6103 . . . . . . . . . . 11 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
87eleq1d 2672 . . . . . . . . . 10 (𝑥 = 𝐴 → ((1st𝑥) ∈ 𝐵 ↔ (1st𝐴) ∈ 𝐵))
9 fveq2 6103 . . . . . . . . . . 11 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
109eleq1d 2672 . . . . . . . . . 10 (𝑥 = 𝐴 → ((2nd𝑥) ∈ 𝐶 ↔ (2nd𝐴) ∈ 𝐶))
118, 10anbi12d 743 . . . . . . . . 9 (𝑥 = 𝐴 → (((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶) ↔ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
126, 11anbi12d 743 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))))
135, 12anbi12d 743 . . . . . . 7 (𝑥 = 𝐴 → (( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))) ↔ ( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))))
1413spcegv 3267 . . . . . 6 (𝐴 ∈ (V × V) → (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → ∃𝑥( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)))))
154, 14mpcom 37 . . . . 5 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → ∃𝑥( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))))
16 eluniab 4383 . . . . 5 ( 𝐴 {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))} ↔ ∃𝑥( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))))
1715, 16sylibr 223 . . . 4 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))})
18 xp2 7094 . . . . . 6 (𝐵 × 𝐶) = {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)}
19 df-rab 2905 . . . . . 6 {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)} = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2018, 19eqtri 2632 . . . . 5 (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2120unieqi 4381 . . . 4 (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2217, 21syl6eleqr 2699 . . 3 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 (𝐵 × 𝐶))
233, 22mpancom 700 . 2 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴 (𝐵 × 𝐶))
241, 23sylbi 206 1 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 (𝐵 × 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173  ∪ cuni 4372   × cxp 5036  ‘cfv 5804  1st c1st 7057  2nd 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)
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