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Theorem unielxp 7443
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 7440 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
2 elvvuni 5386 . . . 4 (𝐴 ∈ (V × V) → 𝐴𝐴)
32adantr 473 . . 3 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴𝐴)
4 simprl 788 . . . . . 6 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 ∈ (V × V))
5 eleq2 2871 . . . . . . . 8 (𝑥 = 𝐴 → ( 𝐴𝑥 𝐴𝐴))
6 eleq1 2870 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 ∈ (V × V) ↔ 𝐴 ∈ (V × V)))
7 fveq2 6415 . . . . . . . . . . 11 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
87eleq1d 2867 . . . . . . . . . 10 (𝑥 = 𝐴 → ((1st𝑥) ∈ 𝐵 ↔ (1st𝐴) ∈ 𝐵))
9 fveq2 6415 . . . . . . . . . . 11 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
109eleq1d 2867 . . . . . . . . . 10 (𝑥 = 𝐴 → ((2nd𝑥) ∈ 𝐶 ↔ (2nd𝐴) ∈ 𝐶))
118, 10anbi12d 625 . . . . . . . . 9 (𝑥 = 𝐴 → (((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶) ↔ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
126, 11anbi12d 625 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))))
135, 12anbi12d 625 . . . . . . 7 (𝑥 = 𝐴 → (( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))) ↔ ( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))))
1413spcegv 3486 . . . . . 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 4643 . . . . 5 ( 𝐴 {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))} ↔ ∃𝑥( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))))
1715, 16sylibr 226 . . . 4 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))})
18 xp2 7442 . . . . . 6 (𝐵 × 𝐶) = {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)}
19 df-rab 3102 . . . . . 6 {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)} = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2018, 19eqtri 2825 . . . . 5 (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2120unieqi 4641 . . . 4 (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2217, 21syl6eleqr 2893 . . 3 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 (𝐵 × 𝐶))
233, 22mpancom 680 . 2 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴 (𝐵 × 𝐶))
241, 23sylbi 209 1 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wex 1875  wcel 2157  {cab 2789  {crab 3097  Vcvv 3389   cuni 4632   × cxp 5314  cfv 6105  1st c1st 7403  2nd c2nd 7404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101  ax-un 7187
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ral 3098  df-rex 3099  df-rab 3102  df-v 3391  df-sbc 3638  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-br 4848  df-opab 4910  df-mpt 4927  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-rn 5327  df-iota 6068  df-fun 6107  df-fv 6113  df-1st 7405  df-2nd 7406
This theorem is referenced by: (None)
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