Theorem fco3 39421

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 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)

Proof of Theorem fco3

Step	Hyp	Ref	Expression
1		fco3.1	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
2		fco3.2	. . . . 5 ⊢ (𝜑 → Fun 𝐺)
3		funco 5928	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
4	1, 2, 3	syl2anc 693	. . . 4 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺))
5		fdmrn 6064	. . . 4 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺))
6	4, 5	sylib 208	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺))
7		dmco 5643	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) = (◡𝐺 “ dom 𝐹)
8	7	feq2i 6037	. . . 4 ⊢ ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺) ↔ (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran (𝐹 ∘ 𝐺))
9	8	a1i 11	. . 3 ⊢ (𝜑 → ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺) ↔ (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran (𝐹 ∘ 𝐺)))
10	6, 9	mpbid 222	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran (𝐹 ∘ 𝐺))
11		rncoss 5386	. . 3 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹
12	11	a1i 11	. 2 ⊢ (𝜑 → ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹)
13	10, 12	fssd 6057	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 196   ⊆ wss 3574  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117   ∘ ccom 5118  Fun wfun 5882  ⟶wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  smfco  41009