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Mirrors > Home > MPE Home > Th. List > rnxpid | Structured version Visualization version GIF version |
Description: The range of a square Cartesian product. (Contributed by FL, 17-May-2010.) |
Ref | Expression |
---|---|
rnxpid | ⊢ ran (𝐴 × 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rn0 5609 | . . 3 ⊢ ran ∅ = ∅ | |
2 | xpeq2 5362 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅)) | |
3 | xp0 5792 | . . . . 5 ⊢ (𝐴 × ∅) = ∅ | |
4 | 2, 3 | syl6eq 2876 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅) |
5 | 4 | rneqd 5584 | . . 3 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅) |
6 | id 22 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
7 | 1, 5, 6 | 3eqtr4a 2886 | . 2 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴) |
8 | rnxp 5804 | . 2 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴) | |
9 | 7, 8 | pm2.61ine 3081 | 1 ⊢ ran (𝐴 × 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∅c0 4143 × cxp 5339 ran crn 5342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rab 3125 df-v 3415 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-br 4873 df-opab 4935 df-xp 5347 df-rel 5348 df-cnv 5349 df-dm 5351 df-rn 5352 |
This theorem is referenced by: sofld 5821 fpwwe2lem13 9778 ustimasn 22401 utopbas 22408 restutop 22410 ovoliunlem1 23667 metideq 30480 poimirlem3 33955 mblfinlem1 33989 rtrclex 38764 |
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