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Theorem fun2ssres 6370
 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 5849 . . . 4 (𝐴 ⊆ dom 𝐺 → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐹𝐴))
21eqcomd 2804 . . 3 (𝐴 ⊆ dom 𝐺 → (𝐹𝐴) = ((𝐹 ↾ dom 𝐺) ↾ 𝐴))
3 funssres 6369 . . . 4 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
43reseq1d 5818 . . 3 ((Fun 𝐹𝐺𝐹) → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐺𝐴))
52, 4sylan9eqr 2855 . 2 (((Fun 𝐹𝐺𝐹) ∧ 𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
653impa 1107 1 ((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ⊆ wss 3881  dom cdm 5520   ↾ cres 5522  Fun wfun 6319 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-res 5532  df-fun 6327 This theorem is referenced by:  wfrlem12  7952  wfrlem14  7954  wfrlem17  7957  tfrlem9  8007  tfrlem9a  8008  tfrlem11  8010  bnj1503  32246  frrlem10  33260  frrlem12  33262
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