|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > xpcomen | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| xpcomen.1 | ⊢ 𝐴 ∈ V | 
| xpcomen.2 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| xpcomen | ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | xpcomen.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 2 | xpcomen.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 3 | 1, 2 | xpex 7773 | . 2 ⊢ (𝐴 × 𝐵) ∈ V | 
| 4 | 2, 1 | xpex 7773 | . 2 ⊢ (𝐵 × 𝐴) ∈ V | 
| 5 | eqid 2737 | . . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) | |
| 6 | 5 | xpcomf1o 9101 | . 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) | 
| 7 | f1oen2g 9009 | . 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 1463 | 1 ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 Vcvv 3480 {csn 4626 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ◡ccnv 5684 –1-1-onto→wf1o 6560 ≈ cen 8982 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1st 8014 df-2nd 8015 df-en 8986 | 
| This theorem is referenced by: xpcomeng 9104 ackbij1lem5 10263 hashxplem 14472 | 
| Copyright terms: Public domain | W3C validator |