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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 4930 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
3 | df-br 4019 | . 2 ⊢ (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷)) | |
4 | df-br 4019 | . . 3 ⊢ (𝐴𝐶𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶) | |
5 | 4 | anbi1i 458 | . 2 ⊢ ((𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
6 | 2, 3, 5 | 3bitr4i 212 | 1 ⊢ (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ (𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∈ wcel 2160 Vcvv 2752 〈cop 3610 class class class wbr 4018 ↾ cres 4646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4650 df-res 4656 |
This theorem is referenced by: dfres2 4977 dfima2 4990 poirr2 5039 cores 5150 resco 5151 rnco 5153 fnres 5351 fvres 5558 nfunsn 5569 1stconst 6247 2ndconst 6248 |
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