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Theorem unielxp 5742
 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 5739 . 2 (A (B × 𝐶) ↔ (A (V × V) ((1stA) B (2ndA) 𝐶)))
2 elvvuni 4347 . . . 4 (A (V × V) → A A)
32adantr 261 . . 3 ((A (V × V) ((1stA) B (2ndA) 𝐶)) → A A)
4 simprl 483 . . . . . 6 (( A A (A (V × V) ((1stA) B (2ndA) 𝐶))) → A (V × V))
5 eleq2 2098 . . . . . . . 8 (x = A → ( A x A A))
6 eleq1 2097 . . . . . . . . 9 (x = A → (x (V × V) ↔ A (V × V)))
7 fveq2 5121 . . . . . . . . . . 11 (x = A → (1stx) = (1stA))
87eleq1d 2103 . . . . . . . . . 10 (x = A → ((1stx) B ↔ (1stA) B))
9 fveq2 5121 . . . . . . . . . . 11 (x = A → (2ndx) = (2ndA))
109eleq1d 2103 . . . . . . . . . 10 (x = A → ((2ndx) 𝐶 ↔ (2ndA) 𝐶))
118, 10anbi12d 442 . . . . . . . . 9 (x = A → (((1stx) B (2ndx) 𝐶) ↔ ((1stA) B (2ndA) 𝐶)))
126, 11anbi12d 442 . . . . . . . 8 (x = A → ((x (V × V) ((1stx) B (2ndx) 𝐶)) ↔ (A (V × V) ((1stA) B (2ndA) 𝐶))))
135, 12anbi12d 442 . . . . . . 7 (x = A → (( A x (x (V × V) ((1stx) B (2ndx) 𝐶))) ↔ ( A A (A (V × V) ((1stA) B (2ndA) 𝐶)))))
1413spcegv 2635 . . . . . 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 3583 . . . . 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 5741 . . . . . 6 (B × 𝐶) = {x (V × V) ∣ ((1stx) B (2ndx) 𝐶)}
19 df-rab 2309 . . . . . 6 {x (V × V) ∣ ((1stx) B (2ndx) 𝐶)} = {x ∣ (x (V × V) ((1stx) B (2ndx) 𝐶))}
2018, 19eqtri 2057 . . . . 5 (B × 𝐶) = {x ∣ (x (V × V) ((1stx) B (2ndx) 𝐶))}
2120unieqi 3581 . . . 4 (B × 𝐶) = {x ∣ (x (V × V) ((1stx) B (2ndx) 𝐶))}
2217, 21syl6eleqr 2128 . . 3 (( A A (A (V × V) ((1stA) B (2ndA) 𝐶))) → A (B × 𝐶))
233, 22mpancom 399 . 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 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  {crab 2304  Vcvv 2551  ∪ cuni 3571   × cxp 4286  ‘cfv 4845  1st c1st 5707  2nd c2nd 5708 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710 This theorem is referenced by: (None)
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