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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 6004 | . 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 2108 Vcvv 3480 class class class wbr 5143 ↾ 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-res 5697 |
| This theorem is referenced by: dfres2 6059 poirr2 6144 cores 6269 resco 6270 rnco 6272 dfpo2 6316 fnres 6695 fvres 6925 nfunsn 6948 eqfunresadj 7380 1stconst 8125 2ndconst 8126 fsplit 8142 fprlem1 8325 wfrlem5OLD 8353 ttrclresv 9757 ttrclselem2 9766 frrlem15 9797 dprd2da 20062 metustid 24567 dvres 25946 dvres2 25947 ltgov 28605 hlimadd 31212 hhcmpl 31219 hhcms 31222 hlim0 31254 dfdm5 35773 dfrn5 35774 txpss3v 35879 brtxp 35881 pprodss4v 35885 brpprod 35886 brimg 35938 brapply 35939 funpartfun 35944 dfrdg4 35952 xrnss3v 38373 funressnfv 47055 funressnvmo 47057 afv2res 47251 tposres0 48777 setrec2lem2 49213 |
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