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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 5696 | . . 3 ⊢ ran 𝑅 = dom ◡𝑅 | |
| 2 | 1 | reseq2i 5994 | . 2 ⊢ (◡𝑅 ↾ ran 𝑅) = (◡𝑅 ↾ dom ◡𝑅) | 
| 3 | relcnv 6122 | . . 3 ⊢ Rel ◡𝑅 | |
| 4 | dfrel5 38347 | . . 3 ⊢ (Rel ◡𝑅 ↔ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅) | |
| 5 | 3, 4 | mpbi 230 | . 2 ⊢ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅 | 
| 6 | 2, 5 | eqtri 2765 | 1 ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ◡ccnv 5684 dom cdm 5685 ran crn 5686 ↾ cres 5687 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 | 
| This theorem is referenced by: alrmomorn 38359 | 
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