Theorem xpcomen 9090

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 7767	. 2 ⊢ (𝐴 × 𝐵) ∈ V
4	2, 1	xpex 7767	. 2 ⊢ (𝐵 × 𝐴) ∈ V
5		eqid 2760	. . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥})
6	5	xpcomf1o 9088	. 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
7		f1oen2g 8998	. 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 1490	1 ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)

This proof depends on syntax axioms:   ∈ wcel 2144  Vcvv 3450  {csn 4584  ∪ cuni 4866   class class class wbr 5102   ↦ cmpt 5185   × cxp 5648  ^◡ccnv 5649  –1-1-onto→wf1o 6537   ≈ cen 8973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-1st 8001  df-2nd 8002  df-en 8977

This theorem is used by:  xpcomeng  9091  ackbij1lem5  10304  hashxplem  14581