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Theorem fssres2 6536
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 6534 . 2 (((𝐹𝐴):𝐴𝐵𝐶𝐴) → ((𝐹𝐴) ↾ 𝐶):𝐶𝐵)
2 resabs1 5870 . . . 4 (𝐶𝐴 → ((𝐹𝐴) ↾ 𝐶) = (𝐹𝐶))
32feq1d 6488 . . 3 (𝐶𝐴 → (((𝐹𝐴) ↾ 𝐶):𝐶𝐵 ↔ (𝐹𝐶):𝐶𝐵))
43adantl 485 . 2 (((𝐹𝐴):𝐴𝐵𝐶𝐴) → (((𝐹𝐴) ↾ 𝐶):𝐶𝐵 ↔ (𝐹𝐶):𝐶𝐵))
51, 4mpbid 235 1 (((𝐹𝐴):𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wss 3919  cres 5544  wf 6339
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-fun 6345  df-fn 6346  df-f 6347
This theorem is referenced by:  efcvx  25047  filnetlem4  33786
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