dmfcoafv - Mathbox for Alexander van der Vekens

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Theorem dmfcoafv 39904

Description: Domains of a function composition, analogous to dmfco 6182. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

**Assertion**
Ref	Expression
dmfcoafv	⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))

**Proof of Theorem dmfcoafv**
Step	Hyp	Ref	Expression
1		dmfco 6182	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝐴) ∈ dom 𝐹))
2		funres 5843	. . . . . . 7 ⊢ (Fun 𝐺 → Fun (𝐺 ↾ {𝐴}))
3	2	anim2i 591	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐺 ∧ Fun 𝐺) → (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
4	3	ancoms 468	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
5		df-dfat 39845	. . . . . 6 ⊢ (𝐺 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
6		afvfundmfveq 39867	. . . . . 6 ⊢ (𝐺 defAt 𝐴 → (𝐺'''𝐴) = (𝐺‘𝐴))
7	5, 6	sylbir 224	. . . . 5 ⊢ ((𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})) → (𝐺'''𝐴) = (𝐺‘𝐴))
8	4, 7	syl 17	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐺'''𝐴) = (𝐺‘𝐴))
9	8	eqcomd 2616	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐺‘𝐴) = (𝐺'''𝐴))
10	9	eleq1d 2672	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐺‘𝐴) ∈ dom 𝐹 ↔ (𝐺'''𝐴) ∈ dom 𝐹))
11	1, 10	bitrd 267	1 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 dom cdm 5038 ↾ cres 5040 ∘ ccom 5042 Fun wfun 5798 ‘cfv 5804 defAt wdfat 39842 '''cafv 39843

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-dfat 39845 df-afv 39846

This theorem is referenced by: (None)

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