MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpcomeng Structured version   Visualization version   GIF version

Theorem xpcomeng 9015
Description: Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
Assertion
Ref Expression
xpcomeng ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))

Proof of Theorem xpcomeng
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpeq1 5652 . . 3 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
2 xpeq2 5659 . . 3 (𝑥 = 𝐴 → (𝑦 × 𝑥) = (𝑦 × 𝐴))
31, 2breq12d 5123 . 2 (𝑥 = 𝐴 → ((𝑥 × 𝑦) ≈ (𝑦 × 𝑥) ↔ (𝐴 × 𝑦) ≈ (𝑦 × 𝐴)))
4 xpeq2 5659 . . 3 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
5 xpeq1 5652 . . 3 (𝑦 = 𝐵 → (𝑦 × 𝐴) = (𝐵 × 𝐴))
64, 5breq12d 5123 . 2 (𝑦 = 𝐵 → ((𝐴 × 𝑦) ≈ (𝑦 × 𝐴) ↔ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)))
7 vex 3452 . . 3 𝑥 ∈ V
8 vex 3452 . . 3 𝑦 ∈ V
97, 8xpcomen 9014 . 2 (𝑥 × 𝑦) ≈ (𝑦 × 𝑥)
103, 6, 9vtocl2g 3534 1 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107   class class class wbr 5110   × cxp 5636  cen 8887
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-1st 7926  df-2nd 7927  df-en 8891
This theorem is referenced by:  xpsnen2g  9016  xpdom1g  9020  omxpen  9025  xpfir  9217  pwdju1  10133  infxp  10158  infmap2  10161  enrelmap  42343  enrelmapr  42344
  Copyright terms: Public domain W3C validator