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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 5760 | . . 3 ⊢ ran ∅ = ∅ | |
2 | xpeq2 5540 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅)) | |
3 | xp0 5982 | . . . . 5 ⊢ (𝐴 × ∅) = ∅ | |
4 | 2, 3 | eqtrdi 2849 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅) |
5 | 4 | rneqd 5772 | . . 3 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅) |
6 | id 22 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
7 | 1, 5, 6 | 3eqtr4a 2859 | . 2 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴) |
8 | rnxp 5994 | . 2 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴) | |
9 | 7, 8 | pm2.61ine 3070 | 1 ⊢ ran (𝐴 × 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∅c0 4243 × cxp 5517 ran crn 5520 |
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-pr 5295 |
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-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: sofld 6011 fpwwe2lem13 10053 ustimasn 22834 utopbas 22841 restutop 22843 ovoliunlem1 24106 metideq 31246 poimirlem3 35060 mblfinlem1 35094 rtrclex 40317 |
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