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Theorem fnssresb 4954
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 4848 . 2 ((𝐹B) Fn B ↔ (Fun (𝐹B) dom (𝐹B) = B))
2 fnfun 4939 . . . . 5 (𝐹 Fn A → Fun 𝐹)
3 funres 4884 . . . . 5 (Fun 𝐹 → Fun (𝐹B))
42, 3syl 14 . . . 4 (𝐹 Fn A → Fun (𝐹B))
54biantrurd 289 . . 3 (𝐹 Fn A → (dom (𝐹B) = B ↔ (Fun (𝐹B) dom (𝐹B) = B)))
6 ssdmres 4576 . . . 4 (B ⊆ dom 𝐹 ↔ dom (𝐹B) = B)
7 fndm 4941 . . . . 5 (𝐹 Fn A → dom 𝐹 = A)
87sseq2d 2967 . . . 4 (𝐹 Fn A → (B ⊆ dom 𝐹BA))
96, 8syl5bbr 183 . . 3 (𝐹 Fn A → (dom (𝐹B) = BBA))
105, 9bitr3d 179 . 2 (𝐹 Fn A → ((Fun (𝐹B) dom (𝐹B) = B) ↔ BA))
111, 10syl5bb 181 1 (𝐹 Fn A → ((𝐹B) Fn BBA))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wss 2911  dom cdm 4288  cres 4290  Fun wfun 4839   Fn wfn 4840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-fun 4847  df-fn 4848
This theorem is referenced by:  fnssres  4955
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