fvco4i - Metamath Proof Explorer

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Theorem fvco4i 6992

Description: Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.)

**Hypotheses**
Ref	Expression
fvco4i.a	⊢ ∅ = (𝐹‘∅)
fvco4i.b	⊢ Fun 𝐺

**Assertion**
Ref	Expression
fvco4i	⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))

**Proof of Theorem fvco4i**
Step	Hyp	Ref	Expression
1		fvco4i.b	. . . 4 ⊢ Fun 𝐺
2		funfn 6578	. . . 4 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺)
3	1, 2	mpbi 229	. . 3 ⊢ 𝐺 Fn dom 𝐺
4		fvco2 6988	. . 3 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝑋 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
5	3, 4	mpan 688	. 2 ⊢ (𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
6		fvco4i.a	. . 3 ⊢ ∅ = (𝐹‘∅)
7		dmcoss 5970	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺
8	7	sseli 3978	. . . 4 ⊢ (𝑋 ∈ dom (𝐹 ∘ 𝐺) → 𝑋 ∈ dom 𝐺)
9		ndmfv 6926	. . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = ∅)
10	8, 9	nsyl5 159	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = ∅)
11		ndmfv 6926	. . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺‘𝑋) = ∅)
12	11	fveq2d 6895	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺‘𝑋)) = (𝐹‘∅))
13	6, 10, 12	3eqtr4a 2798	. 2 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
14	5, 13	pm2.61i 182	1 ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 ∅c0 4322 dom cdm 5676 ∘ ccom 5680 Fun wfun 6537 Fn wfn 6538 ‘cfv 6543

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551

This theorem is referenced by: lidlval 20813 rspval 20814