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Theorem dmopab3 4491
Description: The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
Assertion
Ref Expression
dmopab3 (x A yφ ↔ dom {⟨x, y⟩ ∣ (x A φ)} = A)
Distinct variable group:   x,y,A
Allowed substitution hints:   φ(x,y)

Proof of Theorem dmopab3
StepHypRef Expression
1 df-ral 2305 . 2 (x A yφx(x Ayφ))
2 pm4.71 369 . . 3 ((x Ayφ) ↔ (x A ↔ (x A yφ)))
32albii 1356 . 2 (x(x Ayφ) ↔ x(x A ↔ (x A yφ)))
4 dmopab 4489 . . . . 5 dom {⟨x, y⟩ ∣ (x A φ)} = {xy(x A φ)}
5 19.42v 1783 . . . . . 6 (y(x A φ) ↔ (x A yφ))
65abbii 2150 . . . . 5 {xy(x A φ)} = {x ∣ (x A yφ)}
74, 6eqtri 2057 . . . 4 dom {⟨x, y⟩ ∣ (x A φ)} = {x ∣ (x A yφ)}
87eqeq1i 2044 . . 3 (dom {⟨x, y⟩ ∣ (x A φ)} = A ↔ {x ∣ (x A yφ)} = A)
9 eqcom 2039 . . 3 (A = {x ∣ (x A yφ)} ↔ {x ∣ (x A yφ)} = A)
10 abeq2 2143 . . 3 (A = {x ∣ (x A yφ)} ↔ x(x A ↔ (x A yφ)))
118, 9, 103bitr2ri 198 . 2 (x(x A ↔ (x A yφ)) ↔ dom {⟨x, y⟩ ∣ (x A φ)} = A)
121, 3, 113bitri 195 1 (x A yφ ↔ dom {⟨x, y⟩ ∣ (x A φ)} = A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wral 2300  {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-bnd 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-ral 2305  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:  dmxpm  4498  fnopabg  4965
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