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Mirrors > Home > ILE Home > Th. List > resopab | GIF version |
Description: Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.) |
Ref | Expression |
---|---|
resopab | ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 4551 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) | |
2 | df-xp 4545 | . . . . . 6 ⊢ (𝐴 × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)} | |
3 | vex 2689 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
4 | 3 | biantru 300 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)) |
5 | 4 | opabbii 3995 | . . . . . 6 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)} |
6 | 2, 5 | eqtr4i 2163 | . . . . 5 ⊢ (𝐴 × V) = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} |
7 | 6 | ineq2i 3274 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴}) |
8 | incom 3268 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴}) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
9 | 7, 8 | eqtri 2160 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
10 | inopab 4671 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
11 | 9, 10 | eqtri 2160 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
12 | 1, 11 | eqtri 2160 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∩ cin 3070 {copab 3988 × cxp 4537 ↾ cres 4541 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-xp 4545 df-rel 4546 df-res 4551 |
This theorem is referenced by: resopab2 4866 opabresid 4872 mptpreima 5032 isarep2 5210 resoprab 5867 df1st2 6116 df2nd2 6117 |
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