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| 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 5213 | . . 3 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 2 | nfopab2 5214 | . . 3 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 3 | 1, 2 | dfdmf 5907 | . 2 ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑥 ∣ ∃𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} | 
| 4 | df-br 5144 | . . . . 5 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
| 5 | opabidw 5529 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
| 6 | 4, 5 | bitri 275 | . . . 4 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) | 
| 7 | 6 | exbii 1848 | . . 3 ⊢ (∃𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∃𝑦𝜑) | 
| 8 | 7 | abbii 2809 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} = {𝑥 ∣ ∃𝑦𝜑} | 
| 9 | 3, 8 | eqtri 2765 | 1 ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 〈cop 4632 class class class wbr 5143 {copab 5205 dom cdm 5685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-dm 5695 | 
| This theorem is referenced by: dmopabelb 5927 dmopabss 5929 dmopab3 5930 mptfnf 6703 opabiotadm 6990 fndmin 7065 dmoprab 7536 zfrep6 7979 hartogslem1 9582 dmttrcl 9761 rankf 9834 dfac3 10161 axdc2lem 10488 shftdm 15110 dfiso2 17816 adjeu 31908 satfdm 35374 fmla0 35387 fmlasuc0 35389 | 
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