dmfcoafv - Mathbox for Alexander van der Vekens

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Theorem dmfcoafv 46555

Description: Domains of a function composition, analogous to dmfco 6994. (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 6994	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝐴) ∈ dom 𝐹))
2		funres 6595	. . . . . . 7 ⊢ (Fun 𝐺 → Fun (𝐺 ↾ {𝐴}))
3	2	anim2i 616	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐺 ∧ Fun 𝐺) → (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
4	3	ancoms 458	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
5		df-dfat 46499	. . . . . 6 ⊢ (𝐺 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
6		afvfundmfveq 46518	. . . . . 6 ⊢ (𝐺 defAt 𝐴 → (𝐺'''𝐴) = (𝐺‘𝐴))
7	5, 6	sylbir 234	. . . . 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 2734	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐺‘𝐴) = (𝐺'''𝐴))
10	9	eleq1d 2814	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐺‘𝐴) ∈ dom 𝐹 ↔ (𝐺'''𝐴) ∈ dom 𝐹))
11	1, 10	bitrd 279	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 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {csn 4629 dom cdm 5678 ↾ cres 5680 ∘ ccom 5682 Fun wfun 6542 ‘cfv 6548 defAt wdfat 46496 '''cafv 46497

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-res 5690 df-iota 6500 df-fun 6550 df-fn 6551 df-fv 6556 df-aiota 46465 df-dfat 46499 df-afv 46500

This theorem is referenced by: (None)

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