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Theorem xpcomen 6689
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 𝐴 ∈ V
xpcomen.2 𝐵 ∈ V
Assertion
Ref Expression
xpcomen (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)

Proof of Theorem xpcomen
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 xpcomen.1 . . 3 𝐴 ∈ V
2 xpcomen.2 . . 3 𝐵 ∈ V
31, 2xpex 4624 . 2 (𝐴 × 𝐵) ∈ V
42, 1xpex 4624 . 2 (𝐵 × 𝐴) ∈ V
5 eqid 2117 . . 3 (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})
65xpcomf1o 6687 . 2 (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
7 f1oen2g 6617 . 2 (((𝐴 × 𝐵) ∈ V ∧ (𝐵 × 𝐴) ∈ V ∧ (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
83, 4, 6, 7mp3an 1300 1 (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)
Colors of variables: wff set class
Syntax hints:  wcel 1465  Vcvv 2660  {csn 3497   cuni 3706   class class class wbr 3899  cmpt 3959   × cxp 4507  ccnv 4508  1-1-ontowf1o 5092  cen 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-en 6603
This theorem is referenced by:  xpcomeng  6690
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