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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 5049 | . 2 ⊢ ran 𝑅 = dom ◡𝑅 | |
2 | ercnv 7650 | . . . 4 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅) | |
3 | 2 | dmeqd 5248 | . . 3 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = dom 𝑅) |
4 | erdm 7639 | . . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
5 | 3, 4 | eqtrd 2644 | . 2 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = 𝐴) |
6 | 1, 5 | syl5eq 2656 | 1 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ◡ccnv 5037 dom cdm 5038 ran crn 5039 Er wer 7626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-er 7629 |
This theorem is referenced by: erssxp 7652 ecss 7675 uniqs2 7696 sylow2a 17857 |
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