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Theorem fssres2 6770
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 6768 . 2 (((𝐹𝐴):𝐴𝐵𝐶𝐴) → ((𝐹𝐴) ↾ 𝐶):𝐶𝐵)
2 resabs1 6016 . . . 4 (𝐶𝐴 → ((𝐹𝐴) ↾ 𝐶) = (𝐹𝐶))
32feq1d 6712 . . 3 (𝐶𝐴 → (((𝐹𝐴) ↾ 𝐶):𝐶𝐵 ↔ (𝐹𝐶):𝐶𝐵))
43adantl 480 . 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 394  wss 3949  cres 5684  wf 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-fun 6555  df-fn 6556  df-f 6557
This theorem is referenced by:  efcvx  26406  filnetlem4  35898
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