![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5610 | . . 3 ⊢ ran ∅ = ∅ | |
2 | xpeq2 5363 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅)) | |
3 | xp0 5793 | . . . . 5 ⊢ (𝐴 × ∅) = ∅ | |
4 | 2, 3 | syl6eq 2877 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅) |
5 | 4 | rneqd 5585 | . . 3 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅) |
6 | id 22 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
7 | 1, 5, 6 | 3eqtr4a 2887 | . 2 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴) |
8 | rnxp 5805 | . 2 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴) | |
9 | 7, 8 | pm2.61ine 3082 | 1 ⊢ ran (𝐴 × 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∅c0 4144 × cxp 5340 ran crn 5343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-xp 5348 df-rel 5349 df-cnv 5350 df-dm 5352 df-rn 5353 |
This theorem is referenced by: sofld 5822 fpwwe2lem13 9779 ustimasn 22402 utopbas 22409 restutop 22411 ovoliunlem1 23668 metideq 30470 poimirlem3 33949 mblfinlem1 33983 rtrclex 38758 |
Copyright terms: Public domain | W3C validator |