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Theorem fnresfnco 44207
Description: Composition of two functions, similar to fnco 6494. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
Assertion
Ref Expression
fnresfnco (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → (𝐹𝐺) Fn 𝐵)

Proof of Theorem fnresfnco
StepHypRef Expression
1 fnfun 6479 . . 3 ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → Fun (𝐹 ↾ ran 𝐺))
2 fnfun 6479 . . 3 (𝐺 Fn 𝐵 → Fun 𝐺)
3 funresfunco 6421 . . 3 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2an 599 . 2 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → Fun (𝐹𝐺))
5 fndm 6481 . . . . . 6 ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → dom (𝐹 ↾ ran 𝐺) = ran 𝐺)
6 dmres 5873 . . . . . . . 8 dom (𝐹 ↾ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
76eqeq1i 2742 . . . . . . 7 (dom (𝐹 ↾ ran 𝐺) = ran 𝐺 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
8 df-ss 3883 . . . . . . 7 (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
97, 8sylbb2 241 . . . . . 6 (dom (𝐹 ↾ ran 𝐺) = ran 𝐺 → ran 𝐺 ⊆ dom 𝐹)
105, 9syl 17 . . . . 5 ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → ran 𝐺 ⊆ dom 𝐹)
1110adantr 484 . . . 4 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → ran 𝐺 ⊆ dom 𝐹)
12 dmcosseq 5842 . . . 4 (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹𝐺) = dom 𝐺)
1311, 12syl 17 . . 3 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → dom (𝐹𝐺) = dom 𝐺)
14 fndm 6481 . . . 4 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
1514adantl 485 . . 3 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → dom 𝐺 = 𝐵)
1613, 15eqtrd 2777 . 2 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → dom (𝐹𝐺) = 𝐵)
17 df-fn 6383 . 2 ((𝐹𝐺) Fn 𝐵 ↔ (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = 𝐵))
184, 16, 17sylanbrc 586 1 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  cin 3865  wss 3866  dom cdm 5551  ran crn 5552  cres 5553  ccom 5555  Fun wfun 6374   Fn wfn 6375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-fun 6382  df-fn 6383
This theorem is referenced by:  funcoressn  44208
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