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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 5786 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | 1 | dmeqi 4710 | . 2 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = dom {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
3 | dmopab 4720 | . 2 ⊢ dom {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
4 | exrot3 1653 | . . . . 5 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | 19.42v 1862 | . . . . . 6 ⊢ (∃𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)) | |
6 | 5 | 2exbii 1570 | . . . . 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 2233 | . . 3 ⊢ {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)} |
9 | df-opab 3960 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧𝜑)} | |
10 | 8, 9 | eqtr4i 2141 | . 2 ⊢ {𝑤 ∣ ∃𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} |
11 | 2, 3, 10 | 3eqtri 2142 | 1 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1316 ∃wex 1453 {cab 2103 〈cop 3500 {copab 3958 dom cdm 4509 {coprab 5743 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-dm 4519 df-oprab 5746 |
This theorem is referenced by: dmoprabss 5821 reldmoprab 5824 fnoprabg 5840 dmaddpq 7155 dmmulpq 7156 |
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