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Theorem funresfunco 41526
Description: Composition of two functions, generalization of funco 5966. (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 5966 . 2 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun ((𝐹 ↾ ran 𝐺) ∘ 𝐺))
2 ssid 3657 . . . . 5 ran 𝐺 ⊆ ran 𝐺
3 cores 5676 . . . . 5 (ran 𝐺 ⊆ ran 𝐺 → ((𝐹 ↾ ran 𝐺) ∘ 𝐺) = (𝐹𝐺))
42, 3ax-mp 5 . . . 4 ((𝐹 ↾ ran 𝐺) ∘ 𝐺) = (𝐹𝐺)
54eqcomi 2660 . . 3 (𝐹𝐺) = ((𝐹 ↾ ran 𝐺) ∘ 𝐺)
65funeqi 5947 . 2 (Fun (𝐹𝐺) ↔ Fun ((𝐹 ↾ ran 𝐺) ∘ 𝐺))
71, 6sylibr 224 1 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wss 3607  ran crn 5144  cres 5145  ccom 5147  Fun wfun 5920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-fun 5928
This theorem is referenced by:  fnresfnco  41527
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