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Theorem xpexgALT 6031
 Description: The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4653 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
xpexgALT

Proof of Theorem xpexgALT
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iunid 3868 . . . 4
21xpeq2i 4560 . . 3
3 xpiundi 4597 . . 3
42, 3eqtr3i 2162 . 2
5 id 19 . . 3
6 fconstmpt 4586 . . . . 5
7 mptexg 5645 . . . . 5
86, 7eqeltrid 2226 . . . 4
98ralrimivw 2506 . . 3
10 iunexg 6017 . . 3
115, 9, 10syl2anr 288 . 2
124, 11eqeltrid 2226 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wcel 1480  wral 2416  cvv 2686  csn 3527  ciun 3813   cmpt 3989   cxp 4537 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131 This theorem is referenced by: (None)
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