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Theorem fnssresb 5310
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 5201 . 2 ((𝐹𝐵) Fn 𝐵 ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵))
2 fnfun 5295 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
3 funres 5239 . . . . 5 (Fun 𝐹 → Fun (𝐹𝐵))
42, 3syl 14 . . . 4 (𝐹 Fn 𝐴 → Fun (𝐹𝐵))
54biantrurd 303 . . 3 (𝐹 Fn 𝐴 → (dom (𝐹𝐵) = 𝐵 ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵)))
6 ssdmres 4913 . . . 4 (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹𝐵) = 𝐵)
7 fndm 5297 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
87sseq2d 3177 . . . 4 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
96, 8bitr3id 193 . . 3 (𝐹 Fn 𝐴 → (dom (𝐹𝐵) = 𝐵𝐵𝐴))
105, 9bitr3d 189 . 2 (𝐹 Fn 𝐴 → ((Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵) ↔ 𝐵𝐴))
111, 10syl5bb 191 1 (𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wss 3121  dom cdm 4611  cres 4613  Fun wfun 5192   Fn wfn 5193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-fun 5200  df-fn 5201
This theorem is referenced by:  fnssres  5311
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