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Theorem fnssresb 6671
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 6545 . 2 ((𝐹𝐵) Fn 𝐵 ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵))
2 fnfun 6648 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
32funresd 6590 . . . 4 (𝐹 Fn 𝐴 → Fun (𝐹𝐵))
43biantrurd 532 . . 3 (𝐹 Fn 𝐴 → (dom (𝐹𝐵) = 𝐵 ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵)))
5 ssdmres 6002 . . . 4 (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹𝐵) = 𝐵)
6 fndm 6651 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76sseq2d 4010 . . . 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 205  wa 395   = wceq 1534  wss 3944  dom cdm 5672  cres 5674  Fun wfun 6536   Fn wfn 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-fun 6544  df-fn 6545
This theorem is referenced by:  fnssres  6672  wrdred1hash  14535  rhmsscrnghm  20587  rngcrescrhm  20606  plyreres  26204  xrge0pluscn  33477  icoreresf  36767  fnbrafvb  46457  rngcrescrhmALTV  47265
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