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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 7756 | . 2 ⊢ (𝐴 × 𝐵) ∈ V |
4 | 2, 1 | xpex 7756 | . 2 ⊢ (𝐵 × 𝐴) ∈ V |
5 | eqid 2725 | . . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) | |
6 | 5 | xpcomf1o 9086 | . 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) |
7 | f1oen2g 8989 | . 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 1457 | 1 ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 Vcvv 3461 {csn 4630 ∪ cuni 4909 class class class wbr 5149 ↦ cmpt 5232 × cxp 5676 ◡ccnv 5677 –1-1-onto→wf1o 6548 ≈ cen 8961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-1st 7994 df-2nd 7995 df-en 8965 |
This theorem is referenced by: xpcomeng 9089 ackbij1lem5 10249 hashxplem 14428 |
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