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Theorem xpcomen 6331
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 4480 . 2 (𝐴 × 𝐵) ∈ V
42, 1xpex 4480 . 2 (𝐵 × 𝐴) ∈ V
5 eqid 2056 . . 3 (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})
65xpcomf1o 6329 . 2 (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
7 f1oen2g 6265 . 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 1243 1 (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)
Colors of variables: wff set class
Syntax hints:  wcel 1409  Vcvv 2574  {csn 3402   cuni 3607   class class class wbr 3791  cmpt 3845   × cxp 4370  ccnv 4371  1-1-ontowf1o 4928  cen 6249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-1st 5794  df-2nd 5795  df-en 6252
This theorem is referenced by:  xpcomeng  6332
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