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Mirrors > Home > ILE Home > Th. List > brres | 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 4750 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
3 | df-br 3868 | . 2 ⊢ (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷)) | |
4 | df-br 3868 | . . 3 ⊢ (𝐴𝐶𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶) | |
5 | 4 | anbi1i 447 | . 2 ⊢ ((𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
6 | 2, 3, 5 | 3bitr4i 211 | 1 ⊢ (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ (𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1445 Vcvv 2633 〈cop 3469 class class class wbr 3867 ↾ cres 4469 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-br 3868 df-opab 3922 df-xp 4473 df-res 4479 |
This theorem is referenced by: dfres2 4797 dfima2 4809 poirr2 4857 cores 4968 resco 4969 rnco 4971 fnres 5164 fvres 5364 nfunsn 5373 1stconst 6024 2ndconst 6025 |
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