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Theorem fnresfnco 45282
Description: Composition of two functions, similar to fnco 6619. (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 6603 . . 3 ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → Fun (𝐹 ↾ ran 𝐺))
2 fnfun 6603 . . 3 (𝐺 Fn 𝐵 → Fun 𝐺)
3 funresfunco 6543 . . 3 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2an 597 . 2 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → Fun (𝐹𝐺))
5 fndm 6606 . . . . . 6 ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → dom (𝐹 ↾ ran 𝐺) = ran 𝐺)
6 dmres 5960 . . . . . . . 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 3928 . . . . . . 7 (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
97, 8sylbb2 237 . . . . . 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 482 . . . 4 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → ran 𝐺 ⊆ dom 𝐹)
12 dmcosseq 5929 . . . 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 6606 . . . 4 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
1514adantl 483 . . 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 6500 . 2 ((𝐹𝐺) Fn 𝐵 ↔ (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = 𝐵))
184, 16, 17sylanbrc 584 1 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  cin 3910  wss 3911  dom cdm 5634  ran crn 5635  cres 5636  ccom 5638  Fun wfun 6491   Fn wfn 6492
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 2708  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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  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  df-fn 6500
This theorem is referenced by:  funcoressn  45283
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