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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 5858 | . 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 209 ∧ wa 399 ∈ wcel 2110 Vcvv 3408 class class class wbr 5053 ↾ cres 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-xp 5557 df-res 5563 |
This theorem is referenced by: dfres2 5909 poirr2 5989 cores 6113 resco 6114 rnco 6116 fnres 6504 fvres 6736 nfunsn 6754 1stconst 7868 2ndconst 7869 fsplit 7885 fsplitOLD 7886 fprlem1 8041 wfrlem5 8059 frrlem15 9373 dprd2da 19429 metustid 23452 dvres 24808 dvres2 24809 ltgov 26688 hlimadd 29274 hhcmpl 29281 hhcms 29284 hlim0 29316 dfpo2 33441 eqfunresadj 33454 dfdm5 33466 dfrn5 33467 ttrclresv 33516 txpss3v 33917 brtxp 33919 pprodss4v 33923 brpprod 33924 brimg 33976 brapply 33977 funpartfun 33982 dfrdg4 33990 xrnss3v 36239 funressnfv 44209 funressnvmo 44211 afv2res 44403 setrec2lem2 46071 |
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