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Theorem xpexg 4395
Description: The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
xpexg ((A 𝑉 B 𝑊) → (A × B) V)

Proof of Theorem xpexg
StepHypRef Expression
1 xpsspw 4393 . 2 (A × B) ⊆ 𝒫 𝒫 (AB)
2 unexg 4144 . . 3 ((A 𝑉 B 𝑊) → (AB) V)
3 pwexg 3924 . . 3 ((AB) V → 𝒫 (AB) V)
4 pwexg 3924 . . 3 (𝒫 (AB) V → 𝒫 𝒫 (AB) V)
52, 3, 43syl 17 . 2 ((A 𝑉 B 𝑊) → 𝒫 𝒫 (AB) V)
6 ssexg 3887 . 2 (((A × B) ⊆ 𝒫 𝒫 (AB) 𝒫 𝒫 (AB) V) → (A × B) V)
71, 5, 6sylancr 393 1 ((A 𝑉 B 𝑊) → (A × B) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  Vcvv 2551  cun 2909  wss 2911  𝒫 cpw 3351   × cxp 4286
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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  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-opab 3810  df-xp 4294
This theorem is referenced by:  xpex  4396  resiexg  4596  cnvexg  4798  coexg  4805  fex2  5002  fabexg  5020  resfunexgALT  5679  cofunexg  5680  fnexALT  5682  opabex3d  5690  opabex3  5691  oprabexd  5696  ofmresex  5706  mpt2exxg  5775  tposexg  5814  erex  6066  xpdom2  6241  xpdom3m  6244
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