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Theorem fssres2 6711
Description: Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
Assertion
Ref Expression
fssres2 (((𝐹𝐴):𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)

Proof of Theorem fssres2
StepHypRef Expression
1 fssres 6709 . 2 (((𝐹𝐴):𝐴𝐵𝐶𝐴) → ((𝐹𝐴) ↾ 𝐶):𝐶𝐵)
2 resabs1 5968 . . . 4 (𝐶𝐴 → ((𝐹𝐴) ↾ 𝐶) = (𝐹𝐶))
32feq1d 6654 . . 3 (𝐶𝐴 → (((𝐹𝐴) ↾ 𝐶):𝐶𝐵 ↔ (𝐹𝐶):𝐶𝐵))
43adantl 483 . 2 (((𝐹𝐴):𝐴𝐵𝐶𝐴) → (((𝐹𝐴) ↾ 𝐶):𝐶𝐵 ↔ (𝐹𝐶):𝐶𝐵))
51, 4mpbid 231 1 (((𝐹𝐴):𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wss 3911  cres 5636  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  efcvx  25811  filnetlem4  34856
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