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Theorem unielxp 5827
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.
StepHypRef Expression
1 elxp7 5824 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
2 elvvuni 4431 . . . 4 (𝐴 ∈ (V × V) → 𝐴𝐴)
32adantr 265 . . 3 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴𝐴)
4 simprl 491 . . . . . 6 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 ∈ (V × V))
5 eleq2 2117 . . . . . . . 8 (𝑥 = 𝐴 → ( 𝐴𝑥 𝐴𝐴))
6 eleq1 2116 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 ∈ (V × V) ↔ 𝐴 ∈ (V × V)))
7 fveq2 5205 . . . . . . . . . . 11 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
87eleq1d 2122 . . . . . . . . . 10 (𝑥 = 𝐴 → ((1st𝑥) ∈ 𝐵 ↔ (1st𝐴) ∈ 𝐵))
9 fveq2 5205 . . . . . . . . . . 11 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
109eleq1d 2122 . . . . . . . . . 10 (𝑥 = 𝐴 → ((2nd𝑥) ∈ 𝐶 ↔ (2nd𝐴) ∈ 𝐶))
118, 10anbi12d 450 . . . . . . . . 9 (𝑥 = 𝐴 → (((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶) ↔ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
126, 11anbi12d 450 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))))
135, 12anbi12d 450 . . . . . . 7 (𝑥 = 𝐴 → (( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))) ↔ ( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))))
1413spcegv 2658 . . . . . 6 (𝐴 ∈ (V × V) → (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → ∃𝑥( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)))))
154, 14mpcom 36 . . . . 5 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → ∃𝑥( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))))
16 eluniab 3619 . . . . 5 ( 𝐴 {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))} ↔ ∃𝑥( 𝐴𝑥 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))))
1715, 16sylibr 141 . . . 4 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))})
18 xp2 5826 . . . . . 6 (𝐵 × 𝐶) = {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)}
19 df-rab 2332 . . . . . 6 {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶)} = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2018, 19eqtri 2076 . . . . 5 (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2120unieqi 3617 . . . 4 (𝐵 × 𝐶) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐵 ∧ (2nd𝑥) ∈ 𝐶))}
2217, 21syl6eleqr 2147 . . 3 (( 𝐴𝐴 ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))) → 𝐴 (𝐵 × 𝐶))
233, 22mpancom 407 . 2 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴 (𝐵 × 𝐶))
241, 23sylbi 118 1 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 (𝐵 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  {cab 2042  {crab 2327  Vcvv 2574   cuni 3607   × cxp 4370  cfv 4929  1st c1st 5792  2nd c2nd 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fo 4935  df-fv 4937  df-1st 5794  df-2nd 5795
This theorem is referenced by: (None)
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