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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 3554 | . 2 ⊢ (𝐴 ⊆ dom 𝐵 ↔ (𝐴 ∩ dom 𝐵) = 𝐴) | |
2 | dmres 5339 | . . 3 ⊢ dom (𝐵 ↾ 𝐴) = (𝐴 ∩ dom 𝐵) | |
3 | 2 | eqeq1i 2615 | . 2 ⊢ (dom (𝐵 ↾ 𝐴) = 𝐴 ↔ (𝐴 ∩ dom 𝐵) = 𝐴) |
4 | 1, 3 | bitr4i 266 | 1 ⊢ (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵 ↾ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∩ cin 3539 ⊆ wss 3540 dom cdm 5038 ↾ cres 5040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 df-res 5050 |
This theorem is referenced by: dmresi 5376 fnssresb 5917 fores 6037 foimacnv 6067 dffv2 6181 sbthlem4 7958 hashimarn 13085 dvres3 23483 c1liplem1 23563 lhop1lem 23580 lhop 23583 usgrares1 25939 usgrafilem1 25940 hhssabloi 27503 hhssnv 27505 hhshsslem1 27508 fresf1o 28815 exidreslem 32846 divrngcl 32926 isdrngo2 32927 dvbdfbdioolem1 38818 fourierdlem48 39047 fourierdlem49 39048 fourierdlem71 39070 fourierdlem73 39072 fourierdlem94 39093 fourierdlem111 39110 fourierdlem112 39111 fourierdlem113 39112 fouriersw 39124 fouriercn 39125 dmvon 39496 trlreslem 40907 |
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