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Theorem funssfv 6176
Description: The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)
Assertion
Ref Expression
funssfv ((Fun 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))

Proof of Theorem funssfv
StepHypRef Expression
1 fvres 6174 . . . 4 (𝐴 ∈ dom 𝐺 → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐹𝐴))
21eqcomd 2627 . . 3 (𝐴 ∈ dom 𝐺 → (𝐹𝐴) = ((𝐹 ↾ dom 𝐺)‘𝐴))
3 funssres 5898 . . . 4 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
43fveq1d 6160 . . 3 ((Fun 𝐹𝐺𝐹) → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐺𝐴))
52, 4sylan9eqr 2677 . 2 (((Fun 𝐹𝐺𝐹) ∧ 𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
653impa 1256 1 ((Fun 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wss 3560  dom cdm 5084  cres 5086  Fun wfun 5851  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-res 5096  df-iota 5820  df-fun 5859  df-fv 5865
This theorem is referenced by:  funsssuppss  7281  wfrlem12  7386  wfrlem14  7388  tfrlem9  7441  tfrlem11  7444  ac6sfi  8164  axdc3lem2  9233  axdc3lem4  9235  imasvscaval  16138  pserdv  24121  subgruhgredgd  26103  subumgredg2  26104  subupgr  26106  sspn  27479  bnj945  30605  bnj1502  30679  bnj545  30726  bnj548  30728  subfacp1lem2a  30923  subfacp1lem2b  30924  subfacp1lem5  30927  cvmliftlem10  31037  cvmliftlem13  31039  frrlem11  31546
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