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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 6379 | . . . 4 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺) | |
3 | 1, 2 | mpbi 232 | . . 3 ⊢ 𝐺 Fn dom 𝐺 |
4 | fvco2 6752 | . . 3 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝑋 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) | |
5 | 3, 4 | mpan 688 | . 2 ⊢ (𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) |
6 | fvco4i.a | . . 3 ⊢ ∅ = (𝐹‘∅) | |
7 | dmcoss 5836 | . . . . . 6 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺 | |
8 | 7 | sseli 3962 | . . . . 5 ⊢ (𝑋 ∈ dom (𝐹 ∘ 𝐺) → 𝑋 ∈ dom 𝐺) |
9 | 8 | con3i 157 | . . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺)) |
10 | ndmfv 6694 | . . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = ∅) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = ∅) |
12 | ndmfv 6694 | . . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺‘𝑋) = ∅) | |
13 | 12 | fveq2d 6668 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺‘𝑋)) = (𝐹‘∅)) |
14 | 6, 11, 13 | 3eqtr4a 2882 | . 2 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) |
15 | 5, 14 | pm2.61i 184 | 1 ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 ∅c0 4290 dom cdm 5549 ∘ ccom 5553 Fun wfun 6343 Fn wfn 6344 ‘cfv 6349 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 |
This theorem is referenced by: lidlval 19958 rspval 19959 |
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