fvco4i - Metamath Proof Explorer

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Theorem fvco4i 6983

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 233	. . 3 ⊢ 𝐺 Fn dom 𝐺
4		fvco2 6978	. . 3 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝑋 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
5	3, 4	mpan 702	. 2 ⊢ (𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
6		fvco4i.a	. . 3 ⊢ ∅ = (𝐹‘∅)
7		dmcoss 5965	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺
8	7	sseli 3933	. . . 4 ⊢ (𝑋 ∈ dom (𝐹 ∘ 𝐺) → 𝑋 ∈ dom 𝐺)
9		ndmfv 6913	. . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = ∅)
10	8, 9	nsyl5 160	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = ∅)
11		ndmfv 6913	. . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺‘𝑋) = ∅)
12	11	fveq2d 6885	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺‘𝑋)) = (𝐹‘∅))
13	6, 10, 12	3eqtr4a 2824	. 2 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
14	5, 13	pm2.61i 184	1 ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1570 ∈ wcel 2143 ∅c0 4286 dom cdm 5661 ∘ ccom 5665 Fun wfun 6530 Fn wfn 6531 ‘cfv 6536

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pr 5404

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-fv 6544

This theorem is referenced by: lidlval 21334 rspval 21335