![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > errn | Structured version Visualization version GIF version |
Description: The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
errn | ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 5686 | . 2 ⊢ ran 𝑅 = dom ◡𝑅 | |
2 | ercnv 8720 | . . . 4 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅) | |
3 | 2 | dmeqd 5903 | . . 3 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = dom 𝑅) |
4 | erdm 8709 | . . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
5 | 3, 4 | eqtrd 2772 | . 2 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = 𝐴) |
6 | 1, 5 | eqtrid 2784 | 1 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ◡ccnv 5674 dom cdm 5675 ran crn 5676 Er wer 8696 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-dm 5685 df-rn 5686 df-er 8699 |
This theorem is referenced by: erssxp 8722 ecss 8745 uniqs2 8769 sylow2a 19481 |
Copyright terms: Public domain | W3C validator |