Theorem xpcomen 8850

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.

Step	Hyp	Ref	Expression
1		xpcomen.1	. . 3 ⊢ 𝐴 ∈ V
2		xpcomen.2	. . 3 ⊢ 𝐵 ∈ V
3	1, 2	xpex 7603	. 2 ⊢ (𝐴 × 𝐵) ∈ V
4	2, 1	xpex 7603	. 2 ⊢ (𝐵 × 𝐴) ∈ V
5		eqid 2738	. . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥})
6	5	xpcomf1o 8848	. 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
7		f1oen2g 8756	. 2 ⊢ (((𝐴 × 𝐵) ∈ V ∧ (𝐵 × 𝐴) ∈ V ∧ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
8	3, 4, 6, 7	mp3an 1460	1 ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)

Syntax hints:   ∈ wcel 2106  Vcvv 3432  {csn 4561  ∪ cuni 4839   class class class wbr 5074   ↦ cmpt 5157   × cxp 5587  ^◡ccnv 5588  –1-1-onto→wf1o 6432   ≈ cen 8730

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-1st 7831  df-2nd 7832  df-en 8734

This theorem is referenced by:  xpcomeng  8851  ackbij1lem5  9980  hashxplem  14148