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Theorem fnresfnco 41527
Description: Composition of two functions, similar to fnco 6037. (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 6026 . . 3 ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → Fun (𝐹 ↾ ran 𝐺))
2 fnfun 6026 . . 3 (𝐺 Fn 𝐵 → Fun 𝐺)
3 funresfunco 41526 . . 3 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2an 493 . 2 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → Fun (𝐹𝐺))
5 fndm 6028 . . . . . 6 ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → dom (𝐹 ↾ ran 𝐺) = ran 𝐺)
6 dmres 5454 . . . . . . . 8 dom (𝐹 ↾ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
76eqeq1i 2656 . . . . . . 7 (dom (𝐹 ↾ ran 𝐺) = ran 𝐺 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
8 df-ss 3621 . . . . . . 7 (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
97, 8sylbb2 228 . . . . . 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 5419 . . . 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 6028 . . . 4 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
1514adantl 481 . . 3 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → dom 𝐺 = 𝐵)
1613, 15eqtrd 2685 . 2 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → dom (𝐹𝐺) = 𝐵)
17 df-fn 5929 . 2 ((𝐹𝐺) Fn 𝐵 ↔ (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = 𝐵))
184, 16, 17sylanbrc 699 1 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  cin 3606  wss 3607  dom cdm 5143  ran crn 5144  cres 5145  ccom 5147  Fun wfun 5920   Fn wfn 5921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-fun 5928  df-fn 5929
This theorem is referenced by:  funcoressn  41528
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