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Theorem funeqd 5140
Description: Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
Hypothesis
Ref Expression
funeqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
funeqd (𝜑 → (Fun 𝐴 ↔ Fun 𝐵))

Proof of Theorem funeqd
StepHypRef Expression
1 funeqd.1 . 2 (𝜑𝐴 = 𝐵)
2 funeq 5138 . 2 (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))
31, 2syl 14 1 (𝜑 → (Fun 𝐴 ↔ Fun 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  Fun wfun 5112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079  df-br 3925  df-opab 3985  df-rel 4541  df-cnv 4542  df-co 4543  df-fun 5120
This theorem is referenced by:  funopg  5152  funsng  5164  funcnvuni  5187  f1eq1  5318  frecuzrdgtclt  10187  shftfn  10589  ennnfonelemfun  11915  ennnfonelemf1  11916  isstruct2im  11954  isstruct2r  11955  structfung  11961  setsfun  11979  setsfun0  11980
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