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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 7772 | . 2 ⊢ (𝐴 × 𝐵) ∈ V |
4 | 2, 1 | xpex 7772 | . 2 ⊢ (𝐵 × 𝐴) ∈ V |
5 | eqid 2735 | . . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) | |
6 | 5 | xpcomf1o 9100 | . 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) |
7 | f1oen2g 9008 | . 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 1460 | 1 ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3478 {csn 4631 ∪ cuni 4912 class class class wbr 5148 ↦ cmpt 5231 × cxp 5687 ◡ccnv 5688 –1-1-onto→wf1o 6562 ≈ cen 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1st 8013 df-2nd 8014 df-en 8985 |
This theorem is referenced by: xpcomeng 9103 ackbij1lem5 10261 hashxplem 14469 |
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