Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fco3 Structured version   Visualization version   GIF version

Theorem fco3 39837
 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 6041 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2anc 696 . . . 4 (𝜑 → Fun (𝐹𝐺))
5 fdmrn 6177 . . . 4 (Fun (𝐹𝐺) ↔ (𝐹𝐺):dom (𝐹𝐺)⟶ran (𝐹𝐺))
64, 5sylib 208 . . 3 (𝜑 → (𝐹𝐺):dom (𝐹𝐺)⟶ran (𝐹𝐺))
7 dmco 5756 . . . . 5 dom (𝐹𝐺) = (𝐺 “ dom 𝐹)
87feq2i 6150 . . . 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 5493 . . 3 ran (𝐹𝐺) ⊆ ran 𝐹
1211a1i 11 . 2 (𝜑 → ran (𝐹𝐺) ⊆ ran 𝐹)
1310, 12fssd 6170 1 (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ⊆ wss 3680  ◡ccnv 5217  dom cdm 5218  ran crn 5219   “ cima 5221   ∘ ccom 5222  Fun wfun 5995  ⟶wf 5997 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-fun 6003  df-fn 6004  df-f 6005 This theorem is referenced by:  smfco  41432
 Copyright terms: Public domain W3C validator