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Theorem xpcomen 8326
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 7228 . 2 (𝐴 × 𝐵) ∈ V
42, 1xpex 7228 . 2 (𝐵 × 𝐴) ∈ V
5 eqid 2825 . . 3 (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})
65xpcomf1o 8324 . 2 (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
7 f1oen2g 8245 . 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 1589 1 (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wcel 2164  Vcvv 3414  {csn 4399   cuni 4660   class class class wbr 4875  cmpt 4954   × cxp 5344  ccnv 5345  1-1-ontowf1o 6126  cen 8225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-1st 7433  df-2nd 7434  df-en 8229
This theorem is referenced by:  xpcomeng  8327  hashxplem  13516
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