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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 | . . 3 ⊢ 𝐵 ∈ V | |
2 | 1 | opelres 5322 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
3 | df-br 4584 | . 2 ⊢ (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷)) | |
4 | df-br 4584 | . . 3 ⊢ (𝐴𝐶𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶) | |
5 | 4 | anbi1i 727 | . 2 ⊢ ((𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
6 | 2, 3, 5 | 3bitr4i 291 | 1 ⊢ (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ (𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 ↾ 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-res 5050 |
This theorem is referenced by: dfres2 5372 dfima2 5387 poirr2 5439 cores 5555 resco 5556 rnco 5558 fnres 5921 fvres 6117 nfunsn 6135 1stconst 7152 2ndconst 7153 fsplit 7169 wfrlem5 7306 dprd2da 18264 metustid 22169 dvres 23481 dvres2 23482 ltgov 25292 axhcompl-zf 27239 hlimadd 27434 hhcmpl 27441 hhcms 27444 hlim0 27476 dfpo2 30898 dfdm5 30921 dfrn5 30922 frrlem5 31028 txpss3v 31155 brtxp 31157 pprodss4v 31161 brpprod 31162 brimg 31214 brapply 31215 funpartfun 31220 dfrdg4 31228 funressnfv 39857 dfdfat2 39860 setrec2lem2 42240 |
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