Theorem funcofd 6743

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 6742	. . 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 6733	. 2 ⊢ ((𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)
6	3, 4, 5	syl2anc 583	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)

Syntax hints:   → wi 4  ^◡ccnv 5668  dom cdm 5669  ran crn 5670   “ cima 5672   ∘ ccom 5673  Fun wfun 6530  ⟶wf 6532

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-fun 6538  df-fn 6539  df-f 6540

This theorem is referenced by:  smfco  46072