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Theorem funresfunco 39854
 Description: Composition of two functions, generalization of funco 5842. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
Assertion
Ref Expression
funresfunco ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))

Proof of Theorem funresfunco
StepHypRef Expression
1 funco 5842 . 2 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun ((𝐹 ↾ ran 𝐺) ∘ 𝐺))
2 ssid 3587 . . . . 5 ran 𝐺 ⊆ ran 𝐺
3 cores 5555 . . . . 5 (ran 𝐺 ⊆ ran 𝐺 → ((𝐹 ↾ ran 𝐺) ∘ 𝐺) = (𝐹𝐺))
42, 3ax-mp 5 . . . 4 ((𝐹 ↾ ran 𝐺) ∘ 𝐺) = (𝐹𝐺)
54eqcomi 2619 . . 3 (𝐹𝐺) = ((𝐹 ↾ ran 𝐺) ∘ 𝐺)
65funeqi 5824 . 2 (Fun (𝐹𝐺) ↔ Fun ((𝐹 ↾ ran 𝐺) ∘ 𝐺))
71, 6sylibr 223 1 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ⊆ wss 3540  ran crn 5039   ↾ cres 5040   ∘ ccom 5042  Fun wfun 5798 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-res 5050  df-fun 5806 This theorem is referenced by:  fnresfnco  39855
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