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Theorem opeldm 4481
 Description: Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)
Hypotheses
Ref Expression
opeldm.1 A V
opeldm.2 B V
Assertion
Ref Expression
opeldm (⟨A, B 𝐶A dom 𝐶)

Proof of Theorem opeldm
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 opeldm.2 . . 3 B V
2 opeq2 3541 . . . 4 (y = B → ⟨A, y⟩ = ⟨A, B⟩)
32eleq1d 2103 . . 3 (y = B → (⟨A, y 𝐶 ↔ ⟨A, B 𝐶))
41, 3spcev 2641 . 2 (⟨A, B 𝐶yA, y 𝐶)
5 opeldm.1 . . 3 A V
65eldm2 4476 . 2 (A dom 𝐶yA, y 𝐶)
74, 6sylibr 137 1 (⟨A, B 𝐶A dom 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370  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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298 This theorem is referenced by:  breldm  4482  elreldm  4503  relssres  4591  iss  4597  imadmrn  4621  dfco2a  4764  funssres  4885  funun  4887  iinerm  6114
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