Theorem xpcomen 7130

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 4893	. 2 ⊢ (𝐴 × 𝐵) ∈ V
4	2, 1	xpex 4893	. 2 ⊢ (𝐵 × 𝐴) ∈ V
5		eqid 2238	. . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥})
6	5	xpcomf1o 7128	. 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
7		f1oen2g 7046	. 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 1378	1 ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)

This proof depends on syntax axioms:   ∈ wcel 2209  Vcvv 2821  {csn 3709  ∪ cuni 3935   class class class wbr 4131   ↦ cmpt 4193   × cxp 4774  ^◡ccnv 4775  –1-1-onto→wf1o 5379   ≈ cen 7025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1st 6378  df-2nd 6379  df-en 7028

This theorem is used by:  xpcomeng  7131