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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 6061 | . 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom ◡◡𝐴) | |
2 | relcnv 5967 | . . 3 ⊢ Rel ◡◡𝐴 | |
3 | resdm 5897 | . . 3 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴 |
5 | dmcnvcnv 5803 | . . 3 ⊢ dom ◡◡𝐴 = dom 𝐴 | |
6 | 5 | reseq2i 5850 | . 2 ⊢ (𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom 𝐴) |
7 | 1, 4, 6 | 3eqtr3ri 2853 | 1 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ◡ccnv 5554 dom cdm 5555 ↾ cres 5557 Rel wrel 5560 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 |
This theorem is referenced by: resdmres 6089 fimacnvinrn 6840 dfrel5 35618 |
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