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| Mirrors > Home > MPE Home > Th. List > ssdmres | Structured version Visualization version GIF version | ||
| Description: A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.) |
| Ref | Expression |
|---|---|
| ssdmres | ⊢ (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵 ↾ 𝐴) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfss2 3969 | . 2 ⊢ (𝐴 ⊆ dom 𝐵 ↔ (𝐴 ∩ dom 𝐵) = 𝐴) | |
| 2 | dmres 6030 | . . 3 ⊢ dom (𝐵 ↾ 𝐴) = (𝐴 ∩ dom 𝐵) | |
| 3 | 2 | eqeq1i 2742 | . 2 ⊢ (dom (𝐵 ↾ 𝐴) = 𝐴 ↔ (𝐴 ∩ dom 𝐵) = 𝐴) |
| 4 | 1, 3 | bitr4i 278 | 1 ⊢ (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵 ↾ 𝐴) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∩ cin 3950 ⊆ wss 3951 dom cdm 5685 ↾ cres 5687 |
| 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-dm 5695 df-res 5697 |
| This theorem is referenced by: dmresi 6070 fnssresb 6690 fores 6830 foimacnv 6865 dffv2 7004 fssrescdmd 7146 sbthlem4 9126 hashres 14477 hashimarn 14479 dvres3 25948 c1liplem1 26035 lhop1lem 26052 lhop 26055 usgrres 29325 vtxdginducedm1lem2 29558 wlkres 29688 trlreslem 29717 cyclnumvtx 29820 hhssabloi 31281 hhssnv 31283 hhshsslem1 31286 fresf1o 32641 fsupprnfi 32701 gsumhashmul 33064 cycpmconjvlem 33161 exidreslem 37884 divrngcl 37964 isdrngo2 37965 n0elqs2 38328 dvbdfbdioolem1 45943 fourierdlem48 46169 fourierdlem49 46170 fourierdlem71 46192 fourierdlem73 46194 fourierdlem94 46215 fourierdlem111 46232 fourierdlem112 46233 fourierdlem113 46234 fouriersw 46246 fouriercn 46247 dmvon 46621 isubgrgrim 47897 |
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