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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 | df-ss 3949 | . 2 ⊢ (𝐴 ⊆ dom 𝐵 ↔ (𝐴 ∩ dom 𝐵) = 𝐴) | |
2 | dmres 5868 | . . 3 ⊢ dom (𝐵 ↾ 𝐴) = (𝐴 ∩ dom 𝐵) | |
3 | 2 | eqeq1i 2823 | . 2 ⊢ (dom (𝐵 ↾ 𝐴) = 𝐴 ↔ (𝐴 ∩ dom 𝐵) = 𝐴) |
4 | 1, 3 | bitr4i 279 | 1 ⊢ (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵 ↾ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∩ cin 3932 ⊆ wss 3933 dom cdm 5548 ↾ cres 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-dm 5558 df-res 5560 |
This theorem is referenced by: dmresi 5914 fnssresb 6462 fores 6593 foimacnv 6625 dffv2 6749 sbthlem4 8618 hashres 13787 hashimarn 13789 dvres3 24438 c1liplem1 24520 lhop1lem 24537 lhop 24540 usgrres 27017 vtxdginducedm1lem2 27249 wlkres 27379 trlreslem 27408 hhssabloi 28966 hhssnv 28968 hhshsslem1 28971 fresf1o 30304 cycpmconjvlem 30710 exidreslem 35036 divrngcl 35116 isdrngo2 35117 n0elqs2 35465 dvbdfbdioolem1 42089 fourierdlem48 42316 fourierdlem49 42317 fourierdlem71 42339 fourierdlem73 42341 fourierdlem94 42362 fourierdlem111 42379 fourierdlem112 42380 fourierdlem113 42381 fouriersw 42393 fouriercn 42394 dmvon 42765 |
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