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Theorem unielxp 8051
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 8048 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
2 elvvuni 5765 . . . 4 (𝐴 ∈ (V × V) → 𝐴𝐴)
32adantr 480 . . 3 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴𝐴)
4 simprl 771 . . . . . 6 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 ∈ (V × V))
5 eleq2 2828 . . . . . . . 8 (𝑥 = 𝐴 → ( 𝐴𝑥 𝐴𝐴))
6 eleq1 2827 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 ∈ (V × V) ↔ 𝐴 ∈ (V × V)))
7 fveq2 6907 . . . . . . . . . . 11 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
87eleq1d 2824 . . . . . . . . . 10 (𝑥 = 𝐴 → ((1st𝑥) ∈ 𝐵 ↔ (1st𝐴) ∈ 𝐵))
9 fveq2 6907 . . . . . . . . . . 11 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
109eleq1d 2824 . . . . . . . . . 10 (𝑥 = 𝐴 → ((2nd𝑥) ∈ 𝐶 ↔ (2nd𝐴) ∈ 𝐶))
118, 10anbi12d 632 . . . . . . . . 9 (𝑥 = 𝐴 → (((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶) ↔ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
126, 11anbi12d 632 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))))
135, 12anbi12d 632 . . . . . . 7 (𝑥 = 𝐴 → (( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))) ↔ ( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))))
1413spcegv 3597 . . . . . 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 4926 . . . . 5 ( 𝐴 {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))} ↔ ∃𝑥( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))))
1715, 16sylibr 234 . . . 4 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))})
18 xp2 8050 . . . . . 6 (𝐵 × 𝐶) = {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)}
19 df-rab 3434 . . . . . 6 {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)} = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2018, 19eqtri 2763 . . . . 5 (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2120unieqi 4924 . . . 4 (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2217, 21eleqtrrdi 2850 . . 3 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 (𝐵 × 𝐶))
233, 22mpancom 688 . 2 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴 (𝐵 × 𝐶))
241, 23sylbi 217 1 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  {crab 3433  Vcvv 3478   cuni 4912   × cxp 5687  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by: (None)
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