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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 7212 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | 1 | dmeqi 5773 | . 2 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = dom {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
3 | dmopab 5784 | . 2 ⊢ dom {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
4 | exrot3 2172 | . . . . 5 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | 19.42v 1954 | . . . . . 6 ⊢ (∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) | |
6 | 5 | 2exbii 1849 | . . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) |
7 | 4, 6 | bitri 277 | . . . 4 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) |
8 | 7 | abbii 2886 | . . 3 ⊢ {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)} |
9 | df-opab 5129 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)} | |
10 | 8, 9 | eqtr4i 2847 | . 2 ⊢ {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} |
11 | 2, 3, 10 | 3eqtri 2848 | 1 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∃wex 1780 {cab 2799 〈cop 4573 {copab 5128 dom cdm 5555 {coprab 7157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-dm 5565 df-oprab 7160 |
This theorem is referenced by: dmoprabss 7256 reldmoprab 7259 fnoprabg 7275 1st2val 7717 2nd2val 7718 joindm 17613 meetdm 17627 dmscut 33272 linedegen 33604 |
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