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Theorem fun2ssres 5239
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 4918 . . . 4 (𝐴 ⊆ dom 𝐺 → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐹𝐴))
21eqcomd 2176 . . 3 (𝐴 ⊆ dom 𝐺 → (𝐹𝐴) = ((𝐹 ↾ dom 𝐺) ↾ 𝐴))
3 funssres 5238 . . . 4 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
43reseq1d 4888 . . 3 ((Fun 𝐹𝐺𝐹) → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐺𝐴))
52, 4sylan9eqr 2225 . 2 (((Fun 𝐹𝐺𝐹) ∧ 𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
653impa 1189 1 ((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 973   = wceq 1348  wss 3121  dom cdm 4609  cres 4611  Fun wfun 5190
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-res 4621  df-fun 5198
This theorem is referenced by:  tfrlem9  6295  tfrlemiubacc  6306  tfr1onlemubacc  6322  tfrcllemubacc  6335
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