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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 7470 | . 2 ⊢ (𝐴 × 𝐵) ∈ V |
4 | 2, 1 | xpex 7470 | . 2 ⊢ (𝐵 × 𝐴) ∈ V |
5 | eqid 2821 | . . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) | |
6 | 5 | xpcomf1o 8600 | . 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) |
7 | f1oen2g 8520 | . 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 2110 Vcvv 3495 {csn 4561 ∪ cuni 4832 class class class wbr 5059 ↦ cmpt 5139 × cxp 5548 ◡ccnv 5549 –1-1-onto→wf1o 6349 ≈ cen 8500 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-1st 7683 df-2nd 7684 df-en 8504 |
This theorem is referenced by: xpcomeng 8603 ackbij1lem5 9640 hashxplem 13788 |
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