fvco4i - Metamath Proof Explorer

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Theorem fvco4i 7023

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 6608	. . . 4 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺)
3	1, 2	mpbi 230	. . 3 ⊢ 𝐺 Fn dom 𝐺
4		fvco2 7019	. . 3 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝑋 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
5	3, 4	mpan 689	. 2 ⊢ (𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
6		fvco4i.a	. . 3 ⊢ ∅ = (𝐹‘∅)
7		dmcoss 5997	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺
8	7	sseli 4004	. . . 4 ⊢ (𝑋 ∈ dom (𝐹 ∘ 𝐺) → 𝑋 ∈ dom 𝐺)
9		ndmfv 6955	. . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = ∅)
10	8, 9	nsyl5 159	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = ∅)
11		ndmfv 6955	. . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺‘𝑋) = ∅)
12	11	fveq2d 6924	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺‘𝑋)) = (𝐹‘∅))
13	6, 10, 12	3eqtr4a 2806	. 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 1537 ∈ wcel 2108 ∅c0 4352 dom cdm 5700 ∘ ccom 5704 Fun wfun 6567 Fn wfn 6568 ‘cfv 6573

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-fv 6581

This theorem is referenced by: lidlval 21243 rspval 21244