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Theorem xpexgALT 6289
 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 4981 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 4138 . . . 4
21xpeq2i 4891 . . 3
3 xpiundi 4924 . . 3
42, 3eqtr3i 2457 . 2
5 id 20 . . 3
6 fconstmpt 4913 . . . . 5
7 mptexg 5957 . . . . 5
86, 7syl5eqel 2519 . . . 4
98ralrimivw 2782 . . 3
10 iunexg 5979 . . 3
115, 9, 10syl2anr 465 . 2
124, 11syl5eqel 2519 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wcel 1725  wral 2697  cvv 2948  csn 3806  ciun 4085   cmpt 4258   cxp 4868 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454
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