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Theorem fun2ssres 6269
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 5764 . . . 4 (𝐴 ⊆ dom 𝐺 → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐹𝐴))
21eqcomd 2801 . . 3 (𝐴 ⊆ dom 𝐺 → (𝐹𝐴) = ((𝐹 ↾ dom 𝐺) ↾ 𝐴))
3 funssres 6268 . . . 4 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
43reseq1d 5733 . . 3 ((Fun 𝐹𝐺𝐹) → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐺𝐴))
52, 4sylan9eqr 2853 . 2 (((Fun 𝐹𝐺𝐹) ∧ 𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
653impa 1103 1 ((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1522  wss 3859  dom cdm 5443  cres 5445  Fun wfun 6219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-res 5455  df-fun 6227
This theorem is referenced by:  wfrlem12  7818  wfrlem14  7820  wfrlem17  7823  tfrlem9  7873  tfrlem9a  7874  tfrlem11  7876  bnj1503  31737  frrlem10  32741  frrlem12  32743
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