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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 5176 | . . 3 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
2 | nfopab2 5177 | . . 3 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
3 | 1, 2 | dfdmf 5853 | . 2 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦} |
4 | df-br 5107 | . . . . 5 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) | |
5 | opabidw 5482 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑) | |
6 | 4, 5 | bitri 275 | . . . 4 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑) |
7 | 6 | exbii 1851 | . . 3 ⊢ (∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃𝑦𝜑) |
8 | 7 | abbii 2803 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦} = {𝑥 ∣ ∃𝑦𝜑} |
9 | 3, 8 | eqtri 2761 | 1 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ⟨cop 4593 class class class wbr 5106 {copab 5168 dom cdm 5634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-dm 5644 |
This theorem is referenced by: dmopabelb 5873 dmopabss 5875 dmopab3 5876 mptfnf 6637 opabiotadm 6924 fndmin 6996 dmoprab 7459 zfrep6 7888 hartogslem1 9483 dmttrcl 9662 rankf 9735 dfac3 10062 axdc2lem 10389 shftdm 14962 dfiso2 17660 adjeu 30873 satfdm 34020 fmla0 34033 fmlasuc0 34035 |
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