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Mirrors > Home > MPE Home > Th. List > brresi | Structured version Visualization version GIF version |
Description: Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.) |
Ref | Expression |
---|---|
opelresi.1 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
brresi | ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelresi.1 | . 2 ⊢ 𝐶 ∈ V | |
2 | brres 6007 | . 2 ⊢ (𝐶 ∈ V → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-res 5701 |
This theorem is referenced by: dfres2 6061 poirr2 6147 cores 6271 resco 6272 rnco 6274 dfpo2 6318 fnres 6696 fvres 6926 nfunsn 6949 eqfunresadj 7380 1stconst 8124 2ndconst 8125 fsplit 8141 fprlem1 8324 wfrlem5OLD 8352 ttrclresv 9755 ttrclselem2 9764 frrlem15 9795 dprd2da 20077 metustid 24583 dvres 25961 dvres2 25962 ltgov 28620 hlimadd 31222 hhcmpl 31229 hhcms 31232 hlim0 31264 dfdm5 35754 dfrn5 35755 txpss3v 35860 brtxp 35862 pprodss4v 35866 brpprod 35867 brimg 35919 brapply 35920 funpartfun 35925 dfrdg4 35933 xrnss3v 38354 funressnfv 46993 funressnvmo 46995 afv2res 47189 setrec2lem2 48925 |
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