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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 7469 | . 2 ⊢ (𝐴 × 𝐵) ∈ V |
4 | 2, 1 | xpex 7469 | . 2 ⊢ (𝐵 × 𝐴) ∈ V |
5 | eqid 2820 | . . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) | |
6 | 5 | xpcomf1o 8599 | . 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}):(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) |
7 | f1oen2g 8519 | . 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 1456 | 1 ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 Vcvv 3491 {csn 4560 ∪ cuni 4831 class class class wbr 5059 ↦ cmpt 5139 × cxp 5546 ◡ccnv 5547 –1-1-onto→wf1o 6347 ≈ cen 8499 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-1st 7682 df-2nd 7683 df-en 8503 |
This theorem is referenced by: xpcomeng 8602 ackbij1lem5 9639 hashxplem 13791 |
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