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Mirrors > Home > MPE Home > Th. List > xpid11 | Structured version Visualization version GIF version |
Description: The Cartesian square is a one-to-one construction. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
xpid11 | ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 5906 | . . 3 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) → dom (𝐴 × 𝐴) = dom (𝐵 × 𝐵)) | |
2 | dmxpid 5932 | . . 3 ⊢ dom (𝐴 × 𝐴) = 𝐴 | |
3 | dmxpid 5932 | . . 3 ⊢ dom (𝐵 × 𝐵) = 𝐵 | |
4 | 1, 2, 3 | 3eqtr3g 2791 | . 2 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) → 𝐴 = 𝐵) |
5 | xpeq12 5703 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) → (𝐴 × 𝐴) = (𝐵 × 𝐵)) | |
6 | 5 | anidms 566 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐴) = (𝐵 × 𝐵)) |
7 | 4, 6 | impbii 208 | 1 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 × cxp 5676 dom cdm 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-dm 5688 |
This theorem is referenced by: intopsn 18614 grporn 30344 ismndo2 37347 rngosn3 37397 rngomndo 37408 |
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