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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 5938 | . . 3 ⊢ ran ∅ = ∅ | |
2 | xpeq2 5709 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅)) | |
3 | xp0 6179 | . . . . 5 ⊢ (𝐴 × ∅) = ∅ | |
4 | 2, 3 | eqtrdi 2790 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅) |
5 | 4 | rneqd 5951 | . . 3 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅) |
6 | id 22 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
7 | 1, 5, 6 | 3eqtr4a 2800 | . 2 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴) |
8 | rnxp 6191 | . 2 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴) | |
9 | 7, 8 | pm2.61ine 3022 | 1 ⊢ ran (𝐴 × 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∅c0 4338 × cxp 5686 ran crn 5689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-dm 5698 df-rn 5699 |
This theorem is referenced by: sofld 6208 fpwwe2lem12 10679 ustimasn 24252 utopbas 24259 restutop 24261 ovoliunlem1 25550 metideq 33853 poimirlem3 37609 mblfinlem1 37643 rtrclex 43606 |
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