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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 5494 | . . 3 ⊢ {〈〈x, y〉, z〉 ∣ φ} = {〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} | |
2 | 1 | dmeqi 4479 | . 2 ⊢ dom {〈〈x, y〉, z〉 ∣ φ} = dom {〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} |
3 | dmopab 4489 | . 2 ⊢ dom {〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} = {w ∣ ∃z∃x∃y(w = 〈x, y〉 ∧ φ)} | |
4 | exrot3 1577 | . . . . 5 ⊢ (∃z∃x∃y(w = 〈x, y〉 ∧ φ) ↔ ∃x∃y∃z(w = 〈x, y〉 ∧ φ)) | |
5 | 19.42v 1783 | . . . . . 6 ⊢ (∃z(w = 〈x, y〉 ∧ φ) ↔ (w = 〈x, y〉 ∧ ∃zφ)) | |
6 | 5 | 2exbii 1494 | . . . . 5 ⊢ (∃x∃y∃z(w = 〈x, y〉 ∧ φ) ↔ ∃x∃y(w = 〈x, y〉 ∧ ∃zφ)) |
7 | 4, 6 | bitri 173 | . . . 4 ⊢ (∃z∃x∃y(w = 〈x, y〉 ∧ φ) ↔ ∃x∃y(w = 〈x, y〉 ∧ ∃zφ)) |
8 | 7 | abbii 2150 | . . 3 ⊢ {w ∣ ∃z∃x∃y(w = 〈x, y〉 ∧ φ)} = {w ∣ ∃x∃y(w = 〈x, y〉 ∧ ∃zφ)} |
9 | df-opab 3810 | . . 3 ⊢ {〈x, y〉 ∣ ∃zφ} = {w ∣ ∃x∃y(w = 〈x, y〉 ∧ ∃zφ)} | |
10 | 8, 9 | eqtr4i 2060 | . 2 ⊢ {w ∣ ∃z∃x∃y(w = 〈x, y〉 ∧ φ)} = {〈x, y〉 ∣ ∃zφ} |
11 | 2, 3, 10 | 3eqtri 2061 | 1 ⊢ dom {〈〈x, y〉, z〉 ∣ φ} = {〈x, y〉 ∣ ∃zφ} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1242 ∃wex 1378 {cab 2023 〈cop 3370 {copab 3808 dom cdm 4288 {coprab 5456 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-dm 4298 df-oprab 5459 |
This theorem is referenced by: dmoprabss 5528 reldmoprab 5531 fnoprabg 5544 dmaddpq 6363 dmmulpq 6364 |
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