fvco4i - Metamath Proof Explorer

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Theorem fvco4i 6944

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 6530	. . . 4 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺)
3	1, 2	mpbi 230	. . 3 ⊢ 𝐺 Fn dom 𝐺
4		fvco2 6940	. . 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 5927	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺
8	7	sseli 3939	. . . 4 ⊢ (𝑋 ∈ dom (𝐹 ∘ 𝐺) → 𝑋 ∈ dom 𝐺)
9		ndmfv 6875	. . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = ∅)
10	8, 9	nsyl5 159	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = ∅)
11		ndmfv 6875	. . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺‘𝑋) = ∅)
12	11	fveq2d 6844	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺‘𝑋)) = (𝐹‘∅))
13	6, 10, 12	3eqtr4a 2790	. 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 2109 ∅c0 4292 dom cdm 5631 ∘ ccom 5635 Fun wfun 6493 Fn wfn 6494 ‘cfv 6499

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pr 5382

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-fv 6507

This theorem is referenced by: lidlval 21152 rspval 21153