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Mirrors > Home > ILE Home > Th. List > dmopab | GIF version |
Description: The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.) |
Ref | Expression |
---|---|
dmopab | ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfopab1 3997 | . . 3 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab2 3998 | . . 3 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 1, 2 | dfdmf 4732 | . 2 ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑥 ∣ ∃𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} |
4 | df-br 3930 | . . . . 5 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
5 | opabid 4179 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
6 | 4, 5 | bitri 183 | . . . 4 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
7 | 6 | exbii 1584 | . . 3 ⊢ (∃𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∃𝑦𝜑) |
8 | 7 | abbii 2255 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} = {𝑥 ∣ ∃𝑦𝜑} |
9 | 3, 8 | eqtri 2160 | 1 ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∃wex 1468 ∈ wcel 1480 {cab 2125 〈cop 3530 class class class wbr 3929 {copab 3988 dom cdm 4539 |
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-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 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-br 3930 df-opab 3990 df-dm 4549 |
This theorem is referenced by: dmopabss 4751 dmopab3 4752 fndmin 5527 dmoprab 5852 shftdm 10594 |
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