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Mirrors > Home > MPE Home > Th. List > resopab | Structured version Visualization version 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 5050 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) | |
2 | df-xp 5044 | . . . . . 6 ⊢ (𝐴 × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)} | |
3 | vex 3176 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
4 | 3 | biantru 525 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)) |
5 | 4 | opabbii 4649 | . . . . . 6 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)} |
6 | 2, 5 | eqtr4i 2635 | . . . . 5 ⊢ (𝐴 × V) = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} |
7 | 6 | ineq2i 3773 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴}) |
8 | incom 3767 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴}) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
9 | 7, 8 | eqtri 2632 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
10 | inopab 5174 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
11 | 9, 10 | eqtri 2632 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
12 | 1, 11 | eqtri 2632 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 {copab 4642 × cxp 5036 ↾ cres 5040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 df-rel 5045 df-res 5050 |
This theorem is referenced by: resopab2 5368 opabresid 5374 mptpreima 5545 isarep2 5892 resoprab 6654 elrnmpt2res 6672 df1st2 7150 df2nd2 7151 |
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