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Theorem dmopab3 4475
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 2289 . 2 (x A yφx(x Ayφ))
2 pm4.71 369 . . 3 ((x Ayφ) ↔ (x A ↔ (x A yφ)))
32albii 1339 . 2 (x(x Ayφ) ↔ x(x A ↔ (x A yφ)))
4 dmopab 4473 . . . . 5 dom {⟨x, y⟩ ∣ (x A φ)} = {xy(x A φ)}
5 19.42v 1768 . . . . . 6 (y(x A φ) ↔ (x A yφ))
65abbii 2135 . . . . 5 {xy(x A φ)} = {x ∣ (x A yφ)}
74, 6eqtri 2042 . . . 4 dom {⟨x, y⟩ ∣ (x A φ)} = {x ∣ (x A yφ)}
87eqeq1i 2029 . . 3 (dom {⟨x, y⟩ ∣ (x A φ)} = A ↔ {x ∣ (x A yφ)} = A)
9 eqcom 2024 . . 3 (A = {x ∣ (x A yφ)} ↔ {x ∣ (x A yφ)} = A)
10 abeq2 2128 . . 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 1226   = wceq 1228  wex 1362   wcel 1374  {cab 2008  wral 2284  {copab 3791  dom cdm 4272
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-dm 4282
This theorem is referenced by:  dmxpm  4482  fnopabg  4948
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