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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 7456 | . 2 ⊢ (𝐴 × 𝐵) ∈ V |
4 | 2, 1 | xpex 7456 | . 2 ⊢ (𝐵 × 𝐴) ∈ V |
5 | eqid 2798 | . . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) | |
6 | 5 | xpcomf1o 8589 | . 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) |
7 | f1oen2g 8509 | . 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 1458 | 1 ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 Vcvv 3441 {csn 4525 ∪ cuni 4800 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ◡ccnv 5518 –1-1-onto→wf1o 6323 ≈ cen 8489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-1st 7671 df-2nd 7672 df-en 8493 |
This theorem is referenced by: xpcomeng 8592 ackbij1lem5 9635 hashxplem 13790 |
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