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Mirrors > Home > ILE Home > Th. List > opelres | GIF version |
Description: Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.) |
Ref | Expression |
---|---|
opelres.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opelres | ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 4559 | . . 3 ⊢ (𝐶 ↾ 𝐷) = (𝐶 ∩ (𝐷 × V)) | |
2 | 1 | eleq2i 2207 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ 〈𝐴, 𝐵〉 ∈ (𝐶 ∩ (𝐷 × V))) |
3 | elin 3264 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ∩ (𝐷 × V)) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 〈𝐴, 𝐵〉 ∈ (𝐷 × V))) | |
4 | opelres.1 | . . . 4 ⊢ 𝐵 ∈ V | |
5 | opelxp 4577 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐷 × V) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ V)) | |
6 | 4, 5 | mpbiran2 926 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐷 × V) ↔ 𝐴 ∈ 𝐷) |
7 | 6 | anbi2i 453 | . 2 ⊢ ((〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 〈𝐴, 𝐵〉 ∈ (𝐷 × V)) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
8 | 2, 3, 7 | 3bitri 205 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1481 Vcvv 2689 ∩ cin 3075 〈cop 3535 × cxp 4545 ↾ cres 4549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998 df-xp 4553 df-res 4559 |
This theorem is referenced by: brres 4833 opelresg 4834 opres 4836 dmres 4848 elres 4863 relssres 4865 resiexg 4872 iss 4873 asymref 4932 ssrnres 4989 cnvresima 5036 ressn 5087 funssres 5173 fcnvres 5314 |
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