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Theorem fun2ssres 5969
Description: Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
fun2ssres ((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))

Proof of Theorem fun2ssres
StepHypRef Expression
1 resabs1 5462 . . . 4 (𝐴 ⊆ dom 𝐺 → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐹𝐴))
21eqcomd 2657 . . 3 (𝐴 ⊆ dom 𝐺 → (𝐹𝐴) = ((𝐹 ↾ dom 𝐺) ↾ 𝐴))
3 funssres 5968 . . . 4 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
43reseq1d 5427 . . 3 ((Fun 𝐹𝐺𝐹) → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐺𝐴))
52, 4sylan9eqr 2707 . 2 (((Fun 𝐹𝐺𝐹) ∧ 𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
653impa 1278 1 ((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wss 3607  dom cdm 5143  cres 5145  Fun wfun 5920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-fun 5928
This theorem is referenced by:  wfrlem12  7471  wfrlem14  7473  wfrlem17  7476  tfrlem9  7526  tfrlem9a  7527  tfrlem11  7529  bnj1503  31045  frrlem11  31917
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