Theorem xpcomen 9088

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 7756	. 2 ⊢ (𝐴 × 𝐵) ∈ V
4	2, 1	xpex 7756	. 2 ⊢ (𝐵 × 𝐴) ∈ V
5		eqid 2725	. . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥})
6	5	xpcomf1o 9086	. 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
7		f1oen2g 8989	. 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 1457	1 ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)

Syntax hints:   ∈ wcel 2098  Vcvv 3461  {csn 4630  ∪ cuni 4909   class class class wbr 5149   ↦ cmpt 5232   × cxp 5676  ^◡ccnv 5677  –1-1-onto→wf1o 6548   ≈ cen 8961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-1st 7994  df-2nd 7995  df-en 8965

This theorem is referenced by:  xpcomeng  9089  ackbij1lem5  10249  hashxplem  14428