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

Theorem funresfunco 6543
Description: Composition of two functions, generalization of funco 6542. (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 6542 . 2 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun ((𝐹 ↾ ran 𝐺) ∘ 𝐺))
2 ssid 3967 . . . . 5 ran 𝐺 ⊆ ran 𝐺
3 cores 6202 . . . . 5 (ran 𝐺 ⊆ ran 𝐺 → ((𝐹 ↾ ran 𝐺) ∘ 𝐺) = (𝐹𝐺))
42, 3ax-mp 5 . . . 4 ((𝐹 ↾ ran 𝐺) ∘ 𝐺) = (𝐹𝐺)
54eqcomi 2742 . . 3 (𝐹𝐺) = ((𝐹 ↾ ran 𝐺) ∘ 𝐺)
65funeqi 6523 . 2 (Fun (𝐹𝐺) ↔ Fun ((𝐹 ↾ ran 𝐺) ∘ 𝐺))
71, 6sylibr 233 1 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wss 3911  ran crn 5635  cres 5636  ccom 5638  Fun wfun 6491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-fun 6499
This theorem is referenced by:  fnresfnco  45361
  Copyright terms: Public domain W3C validator