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Theorem fssres2 6375
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 6373 . 2 (((𝐹𝐴):𝐴𝐵𝐶𝐴) → ((𝐹𝐴) ↾ 𝐶):𝐶𝐵)
2 resabs1 5728 . . . 4 (𝐶𝐴 → ((𝐹𝐴) ↾ 𝐶) = (𝐹𝐶))
32feq1d 6329 . . 3 (𝐶𝐴 → (((𝐹𝐴) ↾ 𝐶):𝐶𝐵 ↔ (𝐹𝐶):𝐶𝐵))
43adantl 474 . 2 (((𝐹𝐴):𝐴𝐵𝐶𝐴) → (((𝐹𝐴) ↾ 𝐶):𝐶𝐵 ↔ (𝐹𝐶):𝐶𝐵))
51, 4mpbid 224 1 (((𝐹𝐴):𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wss 3829  cres 5409  wf 6184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-fun 6190  df-fn 6191  df-f 6192
This theorem is referenced by:  efcvx  24740  filnetlem4  33256
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