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Theorem dmoprabss 5528
 Description: The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
dmoprabss dom {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) φ)} ⊆ (A × B)
Distinct variable groups:   x,y,z,A   x,B,y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem dmoprabss
StepHypRef Expression
1 dmoprab 5527 . 2 dom {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) φ)} = {⟨x, y⟩ ∣ z((x A y B) φ)}
2 19.42v 1783 . . . 4 (z((x A y B) φ) ↔ ((x A y B) zφ))
32opabbii 3815 . . 3 {⟨x, y⟩ ∣ z((x A y B) φ)} = {⟨x, y⟩ ∣ ((x A y B) zφ)}
4 opabssxp 4357 . . 3 {⟨x, y⟩ ∣ ((x A y B) zφ)} ⊆ (A × B)
53, 4eqsstri 2969 . 2 {⟨x, y⟩ ∣ z((x A y B) φ)} ⊆ (A × B)
61, 5eqsstri 2969 1 dom {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) φ)} ⊆ (A × B)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911  {copab 3808   × cxp 4286  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-xp 4294  df-dm 4298  df-oprab 5459 This theorem is referenced by:  elmpt2cl  5640  oprabexd  5696  oprabex  5697
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