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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 5569 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) | |
2 | df-xp 5563 | . . . . . 6 ⊢ (𝐴 × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)} | |
3 | vex 3499 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
4 | 3 | biantru 532 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)) |
5 | 4 | opabbii 5135 | . . . . . 6 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)} |
6 | 2, 5 | eqtr4i 2849 | . . . . 5 ⊢ (𝐴 × V) = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} |
7 | 6 | ineq2i 4188 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴}) |
8 | incom 4180 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴}) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
9 | 7, 8 | eqtri 2846 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
10 | inopab 5703 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
11 | 9, 10 | eqtri 2846 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
12 | 1, 11 | eqtri 2846 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∩ cin 3937 {copab 5130 × cxp 5555 ↾ cres 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-opab 5131 df-xp 5563 df-rel 5564 df-res 5569 |
This theorem is referenced by: resopab2 5906 opabresid 5919 opabresidOLD 5921 mptpreima 6094 isarep2 6445 resoprab 7272 elrnmpores 7290 df1st2 7795 df2nd2 7796 imaopab 39126 |
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