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Theorem xpexgALT 5760
 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 4452 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 3712 . . . 4
21xpeq2i 4366 . . 3
3 xpiundi 4398 . . 3
42, 3eqtr3i 2062 . 2
5 id 19 . . 3
6 fconstmpt 4387 . . . . 5
7 mptexg 5386 . . . . 5
86, 7syl5eqel 2124 . . . 4
98ralrimivw 2393 . . 3
10 iunexg 5746 . . 3
115, 9, 10syl2anr 274 . 2
124, 11syl5eqel 2124 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wcel 1393  wral 2306  cvv 2557  csn 3375  ciun 3657   cmpt 3818   cxp 4343 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910 This theorem is referenced by: (None)
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