ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnco Structured version   GIF version

Theorem fnco 4950
Description: Composition of two functions. (Contributed by NM, 22-May-2006.)
Assertion
Ref Expression
fnco ((𝐹 Fn A 𝐺 Fn B ran 𝐺A) → (𝐹𝐺) Fn B)

Proof of Theorem fnco
StepHypRef Expression
1 fnfun 4939 . . . 4 (𝐹 Fn A → Fun 𝐹)
2 fnfun 4939 . . . 4 (𝐺 Fn B → Fun 𝐺)
3 funco 4883 . . . 4 ((Fun 𝐹 Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2an 273 . . 3 ((𝐹 Fn A 𝐺 Fn B) → Fun (𝐹𝐺))
543adant3 923 . 2 ((𝐹 Fn A 𝐺 Fn B ran 𝐺A) → Fun (𝐹𝐺))
6 fndm 4941 . . . . . . 7 (𝐹 Fn A → dom 𝐹 = A)
76sseq2d 2967 . . . . . 6 (𝐹 Fn A → (ran 𝐺 ⊆ dom 𝐹 ↔ ran 𝐺A))
87biimpar 281 . . . . 5 ((𝐹 Fn A ran 𝐺A) → ran 𝐺 ⊆ dom 𝐹)
9 dmcosseq 4546 . . . . 5 (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹𝐺) = dom 𝐺)
108, 9syl 14 . . . 4 ((𝐹 Fn A ran 𝐺A) → dom (𝐹𝐺) = dom 𝐺)
11103adant2 922 . . 3 ((𝐹 Fn A 𝐺 Fn B ran 𝐺A) → dom (𝐹𝐺) = dom 𝐺)
12 fndm 4941 . . . 4 (𝐺 Fn B → dom 𝐺 = B)
13123ad2ant2 925 . . 3 ((𝐹 Fn A 𝐺 Fn B ran 𝐺A) → dom 𝐺 = B)
1411, 13eqtrd 2069 . 2 ((𝐹 Fn A 𝐺 Fn B ran 𝐺A) → dom (𝐹𝐺) = B)
15 df-fn 4848 . 2 ((𝐹𝐺) Fn B ↔ (Fun (𝐹𝐺) dom (𝐹𝐺) = B))
165, 14, 15sylanbrc 394 1 ((𝐹 Fn A 𝐺 Fn B ran 𝐺A) → (𝐹𝐺) Fn B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242  wss 2911  dom cdm 4288  ran crn 4289  ccom 4292  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-eu 1900  df-mo 1901  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-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848
This theorem is referenced by:  fco  4999  fnfco  5008
  Copyright terms: Public domain W3C validator