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Theorem xpmapen 7029
 Description: Equinumerosity law for set exponentiation of a cross product. Exercise 4.47 of [Mendelson] p. 255. (Contributed by NM, 23-Feb-2004.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
Hypotheses
Ref Expression
xpmapen.1
xpmapen.2
xpmapen.3
Assertion
Ref Expression
xpmapen

Proof of Theorem xpmapen
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpmapen.1 . 2
2 xpmapen.2 . 2
3 xpmapen.3 . 2
4 fveq2 5525 . . . 4
54fveq2d 5529 . . 3
65cbvmptv 4111 . 2
74fveq2d 5529 . . 3
87cbvmptv 4111 . 2
9 fveq2 5525 . . . 4
10 fveq2 5525 . . . 4
119, 10opeq12d 3804 . . 3
1211cbvmptv 4111 . 2
131, 2, 3, 6, 8, 12xpmapenlem 7028 1
 Colors of variables: wff set class Syntax hints:   wceq 1623   wcel 1684  cvv 2788  cop 3643   class class class wbr 4023   cmpt 4077   cxp 4687  cfv 5255  (class class class)co 5858  c1st 6120  c2nd 6121   cmap 6772   cen 6860 This theorem is referenced by:  rexpen  12506 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-en 6864
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