Theorem xpcomen 7080

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 4868	. 2 ⊢ (𝐴 × 𝐵) ∈ V
4	2, 1	xpex 4868	. 2 ⊢ (𝐵 × 𝐴) ∈ V
5		eqid 2234	. . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥})
6	5	xpcomf1o 7078	. 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
7		f1oen2g 6996	. 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 1374	1 ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)

Syntax hints:   ∈ wcel 2205  Vcvv 2815  {csn 3691  ∪ cuni 3916   class class class wbr 4111   ↦ cmpt 4173   × cxp 4749  ^◡ccnv 4750  –1-1-onto→wf1o 5353   ≈ cen 6975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-1st 6336  df-2nd 6337  df-en 6978

This theorem is referenced by:  xpcomeng  7081