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Mirrors > Home > ILE Home > Th. List > dmopabss | GIF version |
Description: Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.) |
Ref | Expression |
---|---|
dmopabss | ⊢ dom {〈x, y〉 ∣ (x ∈ A ∧ φ)} ⊆ A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmopab 4489 | . 2 ⊢ dom {〈x, y〉 ∣ (x ∈ A ∧ φ)} = {x ∣ ∃y(x ∈ A ∧ φ)} | |
2 | 19.42v 1783 | . . . 4 ⊢ (∃y(x ∈ A ∧ φ) ↔ (x ∈ A ∧ ∃yφ)) | |
3 | 2 | abbii 2150 | . . 3 ⊢ {x ∣ ∃y(x ∈ A ∧ φ)} = {x ∣ (x ∈ A ∧ ∃yφ)} |
4 | ssab2 3018 | . . 3 ⊢ {x ∣ (x ∈ A ∧ ∃yφ)} ⊆ A | |
5 | 3, 4 | eqsstri 2969 | . 2 ⊢ {x ∣ ∃y(x ∈ A ∧ φ)} ⊆ A |
6 | 1, 5 | eqsstri 2969 | 1 ⊢ dom {〈x, y〉 ∣ (x ∈ A ∧ φ)} ⊆ A |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∃wex 1378 ∈ wcel 1390 {cab 2023 ⊆ wss 2911 {copab 3808 dom cdm 4288 |
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 |
This theorem is referenced by: opabex 5328 |
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