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Mirrors > Home > ILE Home > Th. List > dmoprab | 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 5818 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | 1 | dmeqi 4740 | . 2 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = dom {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
3 | dmopab 4750 | . 2 ⊢ dom {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
4 | exrot3 1668 | . . . . 5 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | 19.42v 1878 | . . . . . 6 ⊢ (∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) | |
6 | 5 | 2exbii 1585 | . . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) |
7 | 4, 6 | bitri 183 | . . . 4 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) |
8 | 7 | abbii 2255 | . . 3 ⊢ {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)} |
9 | df-opab 3990 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)} | |
10 | 8, 9 | eqtr4i 2163 | . 2 ⊢ {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} |
11 | 2, 3, 10 | 3eqtri 2164 | 1 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∃wex 1468 {cab 2125 〈cop 3530 {copab 3988 dom cdm 4539 {coprab 5775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-dm 4549 df-oprab 5778 |
This theorem is referenced by: dmoprabss 5853 reldmoprab 5856 fnoprabg 5872 dmaddpq 7187 dmmulpq 7188 |
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