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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 7247 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | 1 | dmeqi 5758 | . 2 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = dom {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
3 | dmopab 5769 | . 2 ⊢ dom {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
4 | exrot3 2171 | . . . . 5 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | 19.42v 1962 | . . . . . 6 ⊢ (∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) | |
6 | 5 | 2exbii 1856 | . . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) |
7 | 4, 6 | bitri 278 | . . . 4 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) |
8 | 7 | abbii 2801 | . . 3 ⊢ {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)} |
9 | df-opab 5102 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)} | |
10 | 8, 9 | eqtr4i 2762 | . 2 ⊢ {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} |
11 | 2, 3, 10 | 3eqtri 2763 | 1 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∃wex 1787 {cab 2714 〈cop 4533 {copab 5101 dom cdm 5536 {coprab 7192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-dm 5546 df-oprab 7195 |
This theorem is referenced by: dmoprabss 7291 reldmoprab 7294 fnoprabg 7311 1st2val 7767 2nd2val 7768 joindm 17835 meetdm 17849 dmscut 33691 linedegen 34131 |
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