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Theorem xpcomen 9014
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 7692 . 2 (𝐴 × 𝐵) ∈ V
42, 1xpex 7692 . 2 (𝐵 × 𝐴) ∈ V
5 eqid 2737 . . 3 (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})
65xpcomf1o 9012 . 2 (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
7 f1oen2g 8915 . 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 1462 1 (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Vcvv 3448  {csn 4591   cuni 4870   class class class wbr 5110  cmpt 5193   × cxp 5636  ccnv 5637  1-1-ontowf1o 6500  cen 8887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-1st 7926  df-2nd 7927  df-en 8891
This theorem is referenced by:  xpcomeng  9015  ackbij1lem5  10167  hashxplem  14340
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