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Theorem xpcomen 6301
 Description: Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
xpcomen.1
xpcomen.2
Assertion
Ref Expression
xpcomen

Proof of Theorem xpcomen
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 xpcomen.1 . . 3
2 xpcomen.2 . . 3
31, 2xpex 4453 . 2
42, 1xpex 4453 . 2
5 eqid 2040 . . 3
65xpcomf1o 6299 . 2
7 f1oen2g 6235 . 2
83, 4, 6, 7mp3an 1232 1
 Colors of variables: wff set class Syntax hints:   wcel 1393  cvv 2557  csn 3375  cuni 3580   class class class wbr 3764   cmpt 3818   cxp 4343  ccnv 4344  wf1o 4901   cen 6219 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-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-v 2559  df-sbc 2765  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-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-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-1st 5767  df-2nd 5768  df-en 6222 This theorem is referenced by:  xpcomeng  6302
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