![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5531 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) | |
2 | df-xp 5525 | . . . . . 6 ⊢ (𝐴 × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)} | |
3 | vex 3444 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
4 | 3 | biantru 533 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)) |
5 | 4 | opabbii 5097 | . . . . . 6 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)} |
6 | 2, 5 | eqtr4i 2824 | . . . . 5 ⊢ (𝐴 × V) = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} |
7 | 6 | ineq2i 4136 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴}) |
8 | incom 4128 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴}) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
9 | 7, 8 | eqtri 2821 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
10 | inopab 5665 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝐴} ∩ {〈𝑥, 𝑦〉 ∣ 𝜑}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
11 | 9, 10 | eqtri 2821 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ (𝐴 × V)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
12 | 1, 11 | eqtri 2821 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 {copab 5092 × cxp 5517 ↾ cres 5521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 df-rel 5526 df-res 5531 |
This theorem is referenced by: resopab2 5871 opabresid 5884 opabresidOLD 5886 mptpreima 6059 isarep2 6413 resoprab 7249 elrnmpores 7267 df1st2 7776 df2nd2 7777 imaopab 39413 |
Copyright terms: Public domain | W3C validator |