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Theorem funresfunco 6382
 Description: Composition of two functions, generalization of funco 6381. (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 6381 . 2 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun ((𝐹 ↾ ran 𝐺) ∘ 𝐺))
2 ssid 3917 . . . . 5 ran 𝐺 ⊆ ran 𝐺
3 cores 6085 . . . . 5 (ran 𝐺 ⊆ ran 𝐺 → ((𝐹 ↾ ran 𝐺) ∘ 𝐺) = (𝐹𝐺))
42, 3ax-mp 5 . . . 4 ((𝐹 ↾ ran 𝐺) ∘ 𝐺) = (𝐹𝐺)
54eqcomi 2768 . . 3 (𝐹𝐺) = ((𝐹 ↾ ran 𝐺) ∘ 𝐺)
65funeqi 6362 . 2 (Fun (𝐹𝐺) ↔ Fun ((𝐹 ↾ ran 𝐺) ∘ 𝐺))
71, 6sylibr 237 1 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1539   ⊆ wss 3861  ran crn 5530   ↾ cres 5531   ∘ ccom 5533  Fun wfun 6335 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5038  df-opab 5100  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-fun 6343 This theorem is referenced by:  fnresfnco  44045
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