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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 6598 | . . . 4 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺) | |
3 | 1, 2 | mpbi 230 | . . 3 ⊢ 𝐺 Fn dom 𝐺 |
4 | fvco2 7006 | . . 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 5988 | . . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺 | |
8 | 7 | sseli 3991 | . . . 4 ⊢ (𝑋 ∈ dom (𝐹 ∘ 𝐺) → 𝑋 ∈ dom 𝐺) |
9 | ndmfv 6942 | . . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = ∅) | |
10 | 8, 9 | nsyl5 159 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = ∅) |
11 | ndmfv 6942 | . . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺‘𝑋) = ∅) | |
12 | 11 | fveq2d 6911 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺‘𝑋)) = (𝐹‘∅)) |
13 | 6, 10, 12 | 3eqtr4a 2801 | . 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 2106 ∅c0 4339 dom cdm 5689 ∘ ccom 5693 Fun wfun 6557 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: lidlval 21238 rspval 21239 |
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