Theorem funcofd 6761

Description: Composition of two functions as a function with domain and codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof shortened by AV, 20-Sep-2024.)

Hypotheses

Ref	Expression
funcofd.1	⊢ (𝜑 → Fun 𝐹)
funcofd.2	⊢ (𝜑 → Fun 𝐺)

Assertion

Ref	Expression
funcofd	⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)

Proof of Theorem funcofd

Step	Hyp	Ref	Expression
1		funcofd.1	. . 3 ⊢ (𝜑 → Fun 𝐹)
2		fdmrn 6760	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
3	1, 2	sylib 217	. 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ran 𝐹)
4		funcofd.2	. 2 ⊢ (𝜑 → Fun 𝐺)
5		fcof 6751	. 2 ⊢ ((𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)
6	3, 4, 5	syl2anc 582	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)

Syntax hints:   → wi 4  ^◡ccnv 5681  dom cdm 5682  ran crn 5683   “ cima 5685   ∘ ccom 5686  Fun wfun 6547  ⟶wf 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-fun 6555  df-fn 6556  df-f 6557

This theorem is referenced by:  smfco  46219