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Mirrors > Home > MPE Home > Th. List > dmoprab | Structured version Visualization version GIF version |
Description: The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
dmoprab | ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfoprab2 7483 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | 1 | dmeqi 5911 | . 2 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = dom {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
3 | dmopab 5922 | . 2 ⊢ dom {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
4 | exrot3 2155 | . . . . 5 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | 19.42v 1950 | . . . . . 6 ⊢ (∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) | |
6 | 5 | 2exbii 1844 | . . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) |
7 | 4, 6 | bitri 274 | . . . 4 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) |
8 | 7 | abbii 2796 | . . 3 ⊢ {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)} |
9 | df-opab 5216 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)} | |
10 | 8, 9 | eqtr4i 2757 | . 2 ⊢ {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} |
11 | 2, 3, 10 | 3eqtri 2758 | 1 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1534 ∃wex 1774 {cab 2703 〈cop 4639 {copab 5215 dom cdm 5682 {coprab 7425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5154 df-opab 5216 df-dm 5692 df-oprab 7428 |
This theorem is referenced by: dmoprabss 7528 reldmoprab 7531 fnoprabg 7548 1st2val 8031 2nd2val 8032 joindm 18400 meetdm 18414 dmscut 27841 linedegen 35967 |
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