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Theorem fnresfnco 47053
Description: Composition of two functions, similar to fnco 6686. (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 6668 . . 3 ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → Fun (𝐹 ↾ ran 𝐺))
2 fnfun 6668 . . 3 (𝐺 Fn 𝐵 → Fun 𝐺)
3 funresfunco 6607 . . 3 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2an 596 . 2 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → Fun (𝐹𝐺))
5 fndm 6671 . . . . . 6 ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → dom (𝐹 ↾ ran 𝐺) = ran 𝐺)
6 dmres 6030 . . . . . . . 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 dfss2 3969 . . . . . . 7 (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
97, 8sylbb2 238 . . . . . 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 480 . . . 4 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → ran 𝐺 ⊆ dom 𝐹)
12 dmcosseq 5987 . . . 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 6671 . . . 4 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
1514adantl 481 . . 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 6564 . 2 ((𝐹𝐺) Fn 𝐵 ↔ (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = 𝐵))
184, 16, 17sylanbrc 583 1 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  cin 3950  wss 3951  dom cdm 5685  ran crn 5686  cres 5687  ccom 5689  Fun wfun 6555   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-fun 6563  df-fn 6564
This theorem is referenced by:  funcoressn  47054
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