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| Mirrors > Home > MPE Home > Th. List > rnxpid | Structured version Visualization version GIF version | ||
| Description: The range of a Cartesian square. (Contributed by FL, 17-May-2010.) |
| Ref | Expression |
|---|---|
| rnxpid | ⊢ ran (𝐴 × 𝐴) = 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rn0 5936 | . . 3 ⊢ ran ∅ = ∅ | |
| 2 | xpeq2 5706 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅)) | |
| 3 | xp0 6178 | . . . . 5 ⊢ (𝐴 × ∅) = ∅ | |
| 4 | 2, 3 | eqtrdi 2793 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅) |
| 5 | 4 | rneqd 5949 | . . 3 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅) |
| 6 | id 22 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
| 7 | 1, 5, 6 | 3eqtr4a 2803 | . 2 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴) |
| 8 | rnxp 6190 | . 2 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴) | |
| 9 | 7, 8 | pm2.61ine 3025 | 1 ⊢ ran (𝐴 × 𝐴) = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∅c0 4333 × cxp 5683 ran crn 5686 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: sofld 6207 fpwwe2lem12 10682 ustimasn 24237 utopbas 24244 restutop 24246 ovoliunlem1 25537 metideq 33892 poimirlem3 37630 mblfinlem1 37664 rtrclex 43630 |
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