fvco4i - Metamath Proof Explorer

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Theorem fvco4i 6980

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 6566	. . . 4 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺)
3	1, 2	mpbi 230	. . 3 ⊢ 𝐺 Fn dom 𝐺
4		fvco2 6976	. . 3 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝑋 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
5	3, 4	mpan 690	. 2 ⊢ (𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
6		fvco4i.a	. . 3 ⊢ ∅ = (𝐹‘∅)
7		dmcoss 5954	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺
8	7	sseli 3954	. . . 4 ⊢ (𝑋 ∈ dom (𝐹 ∘ 𝐺) → 𝑋 ∈ dom 𝐺)
9		ndmfv 6911	. . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = ∅)
10	8, 9	nsyl5 159	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = ∅)
11		ndmfv 6911	. . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺‘𝑋) = ∅)
12	11	fveq2d 6880	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺‘𝑋)) = (𝐹‘∅))
13	6, 10, 12	3eqtr4a 2796	. 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 1540 ∈ wcel 2108 ∅c0 4308 dom cdm 5654 ∘ ccom 5658 Fun wfun 6525 Fn wfn 6526 ‘cfv 6531

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-fv 6539

This theorem is referenced by: lidlval 21171 rspval 21172