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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvresrn | Structured version Visualization version GIF version |
Description: Converse restricted to range is converse. (Contributed by Peter Mazsa, 3-Sep-2021.) |
Ref | Expression |
---|---|
cnvresrn | ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 5680 | . . 3 ⊢ ran 𝑅 = dom ◡𝑅 | |
2 | 1 | reseq2i 5970 | . 2 ⊢ (◡𝑅 ↾ ran 𝑅) = (◡𝑅 ↾ dom ◡𝑅) |
3 | relcnv 6092 | . . 3 ⊢ Rel ◡𝑅 | |
4 | dfrel5 37020 | . . 3 ⊢ (Rel ◡𝑅 ↔ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅) | |
5 | 3, 4 | mpbi 229 | . 2 ⊢ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅 |
6 | 2, 5 | eqtri 2759 | 1 ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ◡ccnv 5668 dom cdm 5669 ran crn 5670 ↾ cres 5671 Rel wrel 5674 |
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-10 2137 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 |
This theorem is referenced by: alrmomorn 37032 |
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