Theorem funcofd 6556

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 6555	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
3	1, 2	sylib 221	. 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ran 𝐹)
4		funcofd.2	. 2 ⊢ (𝜑 → Fun 𝐺)
5		fcof 6546	. 2 ⊢ ((𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)
6	3, 4, 5	syl2anc 587	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)

Syntax hints:   → wi 4  ^◡ccnv 5535  dom cdm 5536  ran crn 5537   “ cima 5539   ∘ ccom 5540  Fun wfun 6352  ⟶wf 6354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-fun 6360  df-fn 6361  df-f 6362

This theorem is referenced by:  smfco  43951