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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 4610 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) | |
2 | df-xp 4604 | . . . . . 6 ⊢ (𝐴 × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)} | |
3 | vex 2724 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
4 | 3 | biantru 300 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)) |
5 | 4 | opabbii 4043 | . . . . . 6 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)} |
6 | 2, 5 | eqtr4i 2188 | . . . . 5 ⊢ (𝐴 × V) = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} |
7 | 6 | ineq2i 3315 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴}) |
8 | incom 3309 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴}) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
9 | 7, 8 | eqtri 2185 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
10 | inopab 4730 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
11 | 9, 10 | eqtri 2185 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
12 | 1, 11 | eqtri 2185 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∩ cin 3110 {copab 4036 × cxp 4596 ↾ cres 4600 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 df-xp 4604 df-rel 4605 df-res 4610 |
This theorem is referenced by: resopab2 4925 opabresid 4931 mptpreima 5091 isarep2 5269 resoprab 5929 df1st2 6178 df2nd2 6179 |
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