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Theorem fco3 38892
Description: Functionality of a composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
fco3.1 (𝜑 → Fun 𝐹)
fco3.2 (𝜑 → Fun 𝐺)
Assertion
Ref Expression
fco3 (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)

Proof of Theorem fco3
StepHypRef Expression
1 fco3.1 . . . . 5 (𝜑 → Fun 𝐹)
2 fco3.2 . . . . 5 (𝜑 → Fun 𝐺)
3 funco 5886 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2anc 692 . . . 4 (𝜑 → Fun (𝐹𝐺))
5 fdmrn 6021 . . . 4 (Fun (𝐹𝐺) ↔ (𝐹𝐺):dom (𝐹𝐺)⟶ran (𝐹𝐺))
64, 5sylib 208 . . 3 (𝜑 → (𝐹𝐺):dom (𝐹𝐺)⟶ran (𝐹𝐺))
7 dmco 5602 . . . . 5 dom (𝐹𝐺) = (𝐺 “ dom 𝐹)
87feq2i 5994 . . . 4 ((𝐹𝐺):dom (𝐹𝐺)⟶ran (𝐹𝐺) ↔ (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran (𝐹𝐺))
98a1i 11 . . 3 (𝜑 → ((𝐹𝐺):dom (𝐹𝐺)⟶ran (𝐹𝐺) ↔ (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran (𝐹𝐺)))
106, 9mpbid 222 . 2 (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran (𝐹𝐺))
11 rncoss 5346 . . 3 ran (𝐹𝐺) ⊆ ran 𝐹
1211a1i 11 . 2 (𝜑 → ran (𝐹𝐺) ⊆ ran 𝐹)
1310, 12fssd 6014 1 (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wss 3555  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  ccom 5078  Fun wfun 5841  wf 5843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-fun 5849  df-fn 5850  df-f 5851
This theorem is referenced by:  smfco  40313
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