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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 6596 | . . . 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 5985 | . . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺 | |
| 8 | 7 | sseli 3979 | . . . 4 ⊢ (𝑋 ∈ dom (𝐹 ∘ 𝐺) → 𝑋 ∈ dom 𝐺) |
| 9 | ndmfv 6941 | . . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = ∅) | |
| 10 | 8, 9 | nsyl5 159 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = ∅) |
| 11 | ndmfv 6941 | . . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺‘𝑋) = ∅) | |
| 12 | 11 | fveq2d 6910 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺‘𝑋)) = (𝐹‘∅)) |
| 13 | 6, 10, 12 | 3eqtr4a 2803 | . 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 2108 ∅c0 4333 dom cdm 5685 ∘ ccom 5689 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: lidlval 21220 rspval 21221 |
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