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Theorem fnco 5913
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 5902 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fnfun 5902 . . . 4 (𝐺 Fn 𝐵 → Fun 𝐺)
3 funco 5842 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2an 493 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → Fun (𝐹𝐺))
543adant3 1074 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → Fun (𝐹𝐺))
6 fndm 5904 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76sseq2d 3596 . . . . . 6 (𝐹 Fn 𝐴 → (ran 𝐺 ⊆ dom 𝐹 ↔ ran 𝐺𝐴))
87biimpar 501 . . . . 5 ((𝐹 Fn 𝐴 ∧ ran 𝐺𝐴) → ran 𝐺 ⊆ dom 𝐹)
9 dmcosseq 5308 . . . . 5 (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹𝐺) = dom 𝐺)
108, 9syl 17 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = dom 𝐺)
11103adant2 1073 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = dom 𝐺)
12 fndm 5904 . . . 4 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
13123ad2ant2 1076 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom 𝐺 = 𝐵)
1411, 13eqtrd 2644 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = 𝐵)
15 df-fn 5807 . 2 ((𝐹𝐺) Fn 𝐵 ↔ (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = 𝐵))
165, 14, 15sylanbrc 695 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wss 3540  dom cdm 5038  ran crn 5039  ccom 5042  Fun wfun 5798   Fn wfn 5799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807
This theorem is referenced by:  fco  5971  fnfco  5982  fipreima  8155  cshco  13433  swrdco  13434  isofn  16258  prdsinvlem  17347  prdsmgp  18433  pws1  18439  evlslem1  19336  frlmbas  19918  frlmup3  19958  frlmup4  19959  upxp  21236  uptx  21238  0vfval  26845  xppreima2  28830  psgnfzto1stlem  29181  sseqfv1  29778  sseqfn  29779  sseqfv2  29783  volsupnfl  32624  ftc1anclem5  32659  ftc1anclem8  32662  choicefi  38387  fourierdlem42  39042
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