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Theorem fnresfnco 43648
 Description: Composition of two functions, similar to fnco 6437. (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 6423 . . 3 ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → Fun (𝐹 ↾ ran 𝐺))
2 fnfun 6423 . . 3 (𝐺 Fn 𝐵 → Fun 𝐺)
3 funresfunco 6365 . . 3 ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2an 598 . 2 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → Fun (𝐹𝐺))
5 fndm 6425 . . . . . 6 ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → dom (𝐹 ↾ ran 𝐺) = ran 𝐺)
6 dmres 5840 . . . . . . . 8 dom (𝐹 ↾ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
76eqeq1i 2803 . . . . . . 7 (dom (𝐹 ↾ ran 𝐺) = ran 𝐺 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
8 df-ss 3898 . . . . . . 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 5809 . . . 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 6425 . . . 4 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
1514adantl 485 . . 3 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → dom 𝐺 = 𝐵)
1613, 15eqtrd 2833 . 2 (((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → dom (𝐹𝐺) = 𝐵)
17 df-fn 6327 . 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 1538   ∩ cin 3880   ⊆ wss 3881  dom cdm 5519  ran crn 5520   ↾ cres 5521   ∘ ccom 5523  Fun wfun 6318   Fn wfn 6319 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-fun 6326  df-fn 6327 This theorem is referenced by:  funcoressn  43649
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