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Mirrors > Home > MPE Home > Th. List > dmopab | Structured version Visualization version 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 5099 | . . 3 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab2 5100 | . . 3 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 1, 2 | dfdmf 5729 | . 2 ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑥 ∣ ∃𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} |
4 | df-br 5031 | . . . . 5 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
5 | opabidw 5377 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
6 | 4, 5 | bitri 278 | . . . 4 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
7 | 6 | exbii 1849 | . . 3 ⊢ (∃𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∃𝑦𝜑) |
8 | 7 | abbii 2863 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} = {𝑥 ∣ ∃𝑦𝜑} |
9 | 3, 8 | eqtri 2821 | 1 ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 〈cop 4531 class class class wbr 5030 {copab 5092 dom cdm 5519 |
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-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 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-br 5031 df-opab 5093 df-dm 5529 |
This theorem is referenced by: dmopabelb 5749 dmopabss 5751 dmopab3 5752 mptfnf 6455 opabiotadm 6720 fndmin 6792 dmoprab 7234 zfrep6 7638 hartogslem1 8990 rankf 9207 dfac3 9532 axdc2lem 9859 shftdm 14422 dfiso2 17034 adjeu 29672 satfdm 32729 fmla0 32742 fmlasuc0 32744 |
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