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Theorem funresfunco 6273
Description: Composition of two functions, generalization of funco 6272. (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 6272 . 2 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun ((𝐹 ↾ ran 𝐺) ∘ 𝐺))
2 ssid 3916 . . . . 5 ran 𝐺 ⊆ ran 𝐺
3 cores 5984 . . . . 5 (ran 𝐺 ⊆ ran 𝐺 → ((𝐹 ↾ ran 𝐺) ∘ 𝐺) = (𝐹𝐺))
42, 3ax-mp 5 . . . 4 ((𝐹 ↾ ran 𝐺) ∘ 𝐺) = (𝐹𝐺)
54eqcomi 2806 . . 3 (𝐹𝐺) = ((𝐹 ↾ ran 𝐺) ∘ 𝐺)
65funeqi 6253 . 2 (Fun (𝐹𝐺) ↔ Fun ((𝐹 ↾ ran 𝐺) ∘ 𝐺))
71, 6sylibr 235 1 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  wss 3865  ran crn 5451  cres 5452  ccom 5454  Fun wfun 6226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-fun 6234
This theorem is referenced by:  fnresfnco  42814
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