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Theorem unielxp 5723
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 5720 . 2 (A (B × 𝐶) ↔ (A (V × V) ((1stA) B (2ndA) 𝐶)))
2 elvvuni 4331 . . . 4 (A (V × V) → A A)
32adantr 261 . . 3 ((A (V × V) ((1stA) B (2ndA) 𝐶)) → A A)
4 simprl 471 . . . . . 6 (( A A (A (V × V) ((1stA) B (2ndA) 𝐶))) → A (V × V))
5 eleq2 2083 . . . . . . . 8 (x = A → ( A x A A))
6 eleq1 2082 . . . . . . . . 9 (x = A → (x (V × V) ↔ A (V × V)))
7 fveq2 5103 . . . . . . . . . . 11 (x = A → (1stx) = (1stA))
87eleq1d 2088 . . . . . . . . . 10 (x = A → ((1stx) B ↔ (1stA) B))
9 fveq2 5103 . . . . . . . . . . 11 (x = A → (2ndx) = (2ndA))
109eleq1d 2088 . . . . . . . . . 10 (x = A → ((2ndx) 𝐶 ↔ (2ndA) 𝐶))
118, 10anbi12d 445 . . . . . . . . 9 (x = A → (((1stx) B (2ndx) 𝐶) ↔ ((1stA) B (2ndA) 𝐶)))
126, 11anbi12d 445 . . . . . . . 8 (x = A → ((x (V × V) ((1stx) B (2ndx) 𝐶)) ↔ (A (V × V) ((1stA) B (2ndA) 𝐶))))
135, 12anbi12d 445 . . . . . . 7 (x = A → (( A x (x (V × V) ((1stx) B (2ndx) 𝐶))) ↔ ( A A (A (V × V) ((1stA) B (2ndA) 𝐶)))))
1413spcegv 2618 . . . . . 6 (A (V × V) → (( A A (A (V × V) ((1stA) B (2ndA) 𝐶))) → x( A x (x (V × V) ((1stx) B (2ndx) 𝐶)))))
154, 14mpcom 32 . . . . 5 (( A A (A (V × V) ((1stA) B (2ndA) 𝐶))) → x( A x (x (V × V) ((1stx) B (2ndx) 𝐶))))
16 eluniab 3566 . . . . 5 ( A {x ∣ (x (V × V) ((1stx) B (2ndx) 𝐶))} ↔ x( A x (x (V × V) ((1stx) B (2ndx) 𝐶))))
1715, 16sylibr 137 . . . 4 (( A A (A (V × V) ((1stA) B (2ndA) 𝐶))) → A {x ∣ (x (V × V) ((1stx) B (2ndx) 𝐶))})
18 xp2 5722 . . . . . 6 (B × 𝐶) = {x (V × V) ∣ ((1stx) B (2ndx) 𝐶)}
19 df-rab 2293 . . . . . 6 {x (V × V) ∣ ((1stx) B (2ndx) 𝐶)} = {x ∣ (x (V × V) ((1stx) B (2ndx) 𝐶))}
2018, 19eqtri 2042 . . . . 5 (B × 𝐶) = {x ∣ (x (V × V) ((1stx) B (2ndx) 𝐶))}
2120unieqi 3564 . . . 4 (B × 𝐶) = {x ∣ (x (V × V) ((1stx) B (2ndx) 𝐶))}
2217, 21syl6eleqr 2113 . . 3 (( A A (A (V × V) ((1stA) B (2ndA) 𝐶))) → A (B × 𝐶))
233, 22mpancom 401 . 2 ((A (V × V) ((1stA) B (2ndA) 𝐶)) → A (B × 𝐶))
241, 23sylbi 114 1 (A (B × 𝐶) → A (B × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228  wex 1362   wcel 1374  {cab 2008  {crab 2288  Vcvv 2535   cuni 3554   × cxp 4270  cfv 4829  1st c1st 5688  2nd c2nd 5689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-1st 5690  df-2nd 5691
This theorem is referenced by: (None)
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