Theorem fco3OLD 6618

Description: Obsolete version of funcofd 6617 as 20-Sep-2024. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
fco3OLD.1	⊢ (𝜑 → Fun 𝐹)
fco3OLD.2	⊢ (𝜑 → Fun 𝐺)

Assertion

Ref	Expression
fco3OLD	⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)

Proof of Theorem fco3OLD

Step	Hyp	Ref	Expression
1		fco3OLD.1	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
2		fco3OLD.2	. . . . 5 ⊢ (𝜑 → Fun 𝐺)
3		funco 6458	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
4	1, 2, 3	syl2anc 583	. . . 4 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺))
5		fdmrn 6616	. . . 4 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺))
6	4, 5	sylib 217	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺))
7		dmco 6147	. . . 4 ⊢ dom (𝐹 ∘ 𝐺) = (◡𝐺 “ dom 𝐹)
8	7	feq2i 6576	. . 3 ⊢ ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺) ↔ (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran (𝐹 ∘ 𝐺))
9	6, 8	sylib 217	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran (𝐹 ∘ 𝐺))
10		rncoss 5870	. . 3 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹
11	10	a1i 11	. 2 ⊢ (𝜑 → ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹)
12	9, 11	fssd 6602	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)

Syntax hints:   → wi 4   ⊆ wss 3883  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   “ cima 5583   ∘ ccom 5584  Fun wfun 6412  ⟶wf 6414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-fun 6420  df-fn 6421  df-f 6422

This theorem is referenced by: (None)