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Theorem fco3 42210
 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 6368 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2anc 588 . . . 4 (𝜑 → Fun (𝐹𝐺))
5 fdmrn 6516 . . . 4 (Fun (𝐹𝐺) ↔ (𝐹𝐺):dom (𝐹𝐺)⟶ran (𝐹𝐺))
64, 5sylib 221 . . 3 (𝜑 → (𝐹𝐺):dom (𝐹𝐺)⟶ran (𝐹𝐺))
7 dmco 6077 . . . 4 dom (𝐹𝐺) = (𝐺 “ dom 𝐹)
87feq2i 6483 . . 3 ((𝐹𝐺):dom (𝐹𝐺)⟶ran (𝐹𝐺) ↔ (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran (𝐹𝐺))
96, 8sylib 221 . 2 (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran (𝐹𝐺))
10 rncoss 5806 . . 3 ran (𝐹𝐺) ⊆ ran 𝐹
1110a1i 11 . 2 (𝜑 → ran (𝐹𝐺) ⊆ ran 𝐹)
129, 11fssd 6506 1 (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3854  ◡ccnv 5516  dom cdm 5517  ran crn 5518   “ cima 5520   ∘ ccom 5521  Fun wfun 6322  ⟶wf 6324 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pr 5291 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ral 3073  df-rex 3074  df-rab 3077  df-v 3409  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522  df-br 5026  df-opab 5088  df-id 5423  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-fun 6330  df-fn 6331  df-f 6332 This theorem is referenced by:  smfco  43785
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