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Theorem fnssresb 6690
Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
Assertion
Ref Expression
fnssresb (𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))

Proof of Theorem fnssresb
StepHypRef Expression
1 df-fn 6564 . 2 ((𝐹𝐵) Fn 𝐵 ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵))
2 fnfun 6668 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
32funresd 6609 . . . 4 (𝐹 Fn 𝐴 → Fun (𝐹𝐵))
43biantrurd 532 . . 3 (𝐹 Fn 𝐴 → (dom (𝐹𝐵) = 𝐵 ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵)))
5 ssdmres 6031 . . . 4 (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹𝐵) = 𝐵)
6 fndm 6671 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76sseq2d 4016 . . . 4 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
85, 7bitr3id 285 . . 3 (𝐹 Fn 𝐴 → (dom (𝐹𝐵) = 𝐵𝐵𝐴))
94, 8bitr3d 281 . 2 (𝐹 Fn 𝐴 → ((Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵) ↔ 𝐵𝐴))
101, 9bitrid 283 1 (𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wss 3951  dom cdm 5685  cres 5687  Fun wfun 6555   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-fun 6563  df-fn 6564
This theorem is referenced by:  fnssres  6691  wrdred1hash  14599  rhmsscrnghm  20665  rngcrescrhm  20684  plyreres  26324  xrge0pluscn  33939  icoreresf  37353  fnbrafvb  47166  rngcrescrhmALTV  48196
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