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Mirrors > Home > MPE Home > Th. List > resdm2 | Structured version Visualization version GIF version |
Description: A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.) |
Ref | Expression |
---|---|
resdm2 | ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rescnvcnv 6203 | . 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom ◡◡𝐴) | |
2 | relcnv 6103 | . . 3 ⊢ Rel ◡◡𝐴 | |
3 | resdm 6026 | . . 3 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴 |
5 | dmcnvcnv 5932 | . . 3 ⊢ dom ◡◡𝐴 = dom 𝐴 | |
6 | 5 | reseq2i 5978 | . 2 ⊢ (𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom 𝐴) |
7 | 1, 4, 6 | 3eqtr3ri 2768 | 1 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ◡ccnv 5675 dom cdm 5676 ↾ cres 5678 Rel wrel 5681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 |
This theorem is referenced by: resdmres 6231 fimacnvinrn 7073 dfrel5 37679 |
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