Theorem funcofd 6688

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 6687	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
3	1, 2	sylib 218	. 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ran 𝐹)
4		funcofd.2	. 2 ⊢ (𝜑 → Fun 𝐺)
5		fcof 6679	. 2 ⊢ ((𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)
6	3, 4, 5	syl2anc 584	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)

Syntax hints:   → wi 4  ^◡ccnv 5622  dom cdm 5623  ran crn 5624   “ cima 5626   ∘ ccom 5627  Fun wfun 6480  ⟶wf 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-fun 6488  df-fn 6489  df-f 6490

This theorem is referenced by:  smfco  46803