Theorem fco3OLD 6752

Description: Obsolete version of funcofd 6751 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 6589	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
4	1, 2, 3	syl2anc 585	. . . 4 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺))
5		fdmrn 6750	. . . 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 6254	. . . 4 ⊢ dom (𝐹 ∘ 𝐺) = (◡𝐺 “ dom 𝐹)
8	7	feq2i 6710	. . 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 5972	. . 3 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹
11	10	a1i 11	. 2 ⊢ (𝜑 → ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹)
12	9, 11	fssd 6736	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)

Syntax hints:   → wi 4   ⊆ wss 3949  ^◡ccnv 5676  dom cdm 5677  ran crn 5678   “ cima 5680   ∘ ccom 5681  Fun wfun 6538  ⟶wf 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548

This theorem is referenced by: (None)