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Theorem xpcomen 6721
 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 4654 . 2 (𝐴 × 𝐵) ∈ V
42, 1xpex 4654 . 2 (𝐵 × 𝐴) ∈ V
5 eqid 2139 . . 3 (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})
65xpcomf1o 6719 . 2 (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
7 f1oen2g 6649 . 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 1315 1 (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1480  Vcvv 2686  {csn 3527  ∪ cuni 3736   class class class wbr 3929   ↦ cmpt 3989   × cxp 4537  ◡ccnv 4538  –1-1-onto→wf1o 5122   ≈ cen 6632 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-en 6635 This theorem is referenced by:  xpcomeng  6722
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