MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnco Structured version   Visualization version   GIF version

Theorem fnco 5957
Description: Composition of two functions. (Contributed by NM, 22-May-2006.)
Assertion
Ref Expression
fnco ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)

Proof of Theorem fnco
StepHypRef Expression
1 fnfun 5946 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fnfun 5946 . . . 4 (𝐺 Fn 𝐵 → Fun 𝐺)
3 funco 5886 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2an 494 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → Fun (𝐹𝐺))
543adant3 1079 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → Fun (𝐹𝐺))
6 fndm 5948 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76sseq2d 3612 . . . . . 6 (𝐹 Fn 𝐴 → (ran 𝐺 ⊆ dom 𝐹 ↔ ran 𝐺𝐴))
87biimpar 502 . . . . 5 ((𝐹 Fn 𝐴 ∧ ran 𝐺𝐴) → ran 𝐺 ⊆ dom 𝐹)
9 dmcosseq 5347 . . . . 5 (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹𝐺) = dom 𝐺)
108, 9syl 17 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = dom 𝐺)
11103adant2 1078 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = dom 𝐺)
12 fndm 5948 . . . 4 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
13123ad2ant2 1081 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom 𝐺 = 𝐵)
1411, 13eqtrd 2655 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = 𝐵)
15 df-fn 5850 . 2 ((𝐹𝐺) Fn 𝐵 ↔ (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = 𝐵))
165, 14, 15sylanbrc 697 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wss 3555  dom cdm 5074  ran crn 5075  ccom 5078  Fun wfun 5841   Fn wfn 5842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-fun 5849  df-fn 5850
This theorem is referenced by:  fco  6015  fnfco  6026  fipreima  8216  cshco  13519  swrdco  13520  isofn  16356  prdsinvlem  17445  prdsmgp  18531  pws1  18537  evlslem1  19434  frlmbas  20018  frlmup3  20058  frlmup4  20059  upxp  21336  uptx  21338  0vfval  27310  xppreima2  29292  psgnfzto1stlem  29635  sseqfv1  30232  sseqfn  30233  sseqfv2  30237  volsupnfl  33086  ftc1anclem5  33121  ftc1anclem8  33124  choicefi  38866  fourierdlem42  39673
  Copyright terms: Public domain W3C validator