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Theorem xpcomen 6052
 Description: Commutative law for equinumerosity of cross product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by set.mm contributors, 5-Jan-2004.) (Revised by set.mm contributors, 23-Apr-2014.)
Hypotheses
Ref Expression
xpcomen.1 A V
xpcomen.2 B V
Assertion
Ref Expression
xpcomen (A × B) ≈ (B × A)

Proof of Theorem xpcomen
StepHypRef Expression
1 swapres 5512 . . 3 ( Swap (A × B)):(A × B)–1-1-onto(A × B)
2 cnvxp 5043 . . . 4 (A × B) = (B × A)
3 f1oeq3 5283 . . . 4 ((A × B) = (B × A) → (( Swap (A × B)):(A × B)–1-1-onto(A × B) ↔ ( Swap (A × B)):(A × B)–1-1-onto→(B × A)))
42, 3ax-mp 8 . . 3 (( Swap (A × B)):(A × B)–1-1-onto(A × B) ↔ ( Swap (A × B)):(A × B)–1-1-onto→(B × A))
51, 4mpbi 199 . 2 ( Swap (A × B)):(A × B)–1-1-onto→(B × A)
6 swapex 4742 . . . 4 Swap V
7 xpcomen.1 . . . . 5 A V
8 xpcomen.2 . . . . 5 B V
97, 8xpex 5115 . . . 4 (A × B) V
106, 9resex 5117 . . 3 ( Swap (A × B)) V
1110f1oen 6033 . 2 (( Swap (A × B)):(A × B)–1-1-onto→(B × A) → (A × B) ≈ (B × A))
125, 11ax-mp 8 1 (A × B) ≈ (B × A)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2859   class class class wbr 4639   Swap cswap 4718   × cxp 4770  ◡ccnv 4771   ↾ cres 4774  –1-1-onto→wf1o 4780   ≈ cen 6028 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-en 6029 This theorem is referenced by:  xpcomeng  6053  muccom  6134
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