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Mirrors > Home > MPE Home > Th. List > fvco4i | Structured version Visualization version GIF version |
Description: Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Ref | Expression |
---|---|
fvco4i.a | ⊢ ∅ = (𝐹‘∅) |
fvco4i.b | ⊢ Fun 𝐺 |
Ref | Expression |
---|---|
fvco4i | ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvco4i.b | . . . 4 ⊢ Fun 𝐺 | |
2 | funfn 6571 | . . . 4 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺) | |
3 | 1, 2 | mpbi 229 | . . 3 ⊢ 𝐺 Fn dom 𝐺 |
4 | fvco2 6981 | . . 3 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝑋 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) | |
5 | 3, 4 | mpan 687 | . 2 ⊢ (𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) |
6 | fvco4i.a | . . 3 ⊢ ∅ = (𝐹‘∅) | |
7 | dmcoss 5963 | . . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺 | |
8 | 7 | sseli 3973 | . . . 4 ⊢ (𝑋 ∈ dom (𝐹 ∘ 𝐺) → 𝑋 ∈ dom 𝐺) |
9 | ndmfv 6919 | . . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = ∅) | |
10 | 8, 9 | nsyl5 159 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = ∅) |
11 | ndmfv 6919 | . . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺‘𝑋) = ∅) | |
12 | 11 | fveq2d 6888 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺‘𝑋)) = (𝐹‘∅)) |
13 | 6, 10, 12 | 3eqtr4a 2792 | . 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 1533 ∈ wcel 2098 ∅c0 4317 dom cdm 5669 ∘ ccom 5673 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 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-fv 6544 |
This theorem is referenced by: lidlval 21067 rspval 21068 |
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