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Mirrors > Home > MPE Home > Th. List > brres | Structured version Visualization version GIF version |
Description: Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.) |
Ref | Expression |
---|---|
opelres.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brres | ⊢ (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ (𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelres.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | brresg 5435 | . 2 ⊢ (𝐵 ∈ V → (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ (𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ (𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 ∈ wcel 2030 Vcvv 3231 class class class wbr 4685 ↾ cres 5145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-xp 5149 df-res 5155 |
This theorem is referenced by: dfres2 5488 dfima2 5503 poirr2 5555 cores 5676 resco 5677 rnco 5679 fnres 6045 fvres 6245 nfunsn 6263 1stconst 7310 2ndconst 7311 fsplit 7327 wfrlem5 7464 dprd2da 18487 metustid 22406 dvres 23720 dvres2 23721 ltgov 25537 axhcompl-zf 27983 hlimadd 28178 hhcmpl 28185 hhcms 28188 hlim0 28220 dfpo2 31771 eqfunresadj 31785 dfdm5 31800 dfrn5 31801 frrlem5 31909 txpss3v 32110 brtxp 32112 pprodss4v 32116 brpprod 32117 brimg 32169 brapply 32170 funpartfun 32175 dfrdg4 32183 xrnss3v 34274 funressnfv 41529 dfdfat2 41532 setrec2lem2 42766 |
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