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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 6532 | . . . 4 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺) | |
3 | 1, 2 | mpbi 229 | . . 3 ⊢ 𝐺 Fn dom 𝐺 |
4 | fvco2 6939 | . . 3 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝑋 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) | |
5 | 3, 4 | mpan 689 | . 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 3941 | . . . 4 ⊢ (𝑋 ∈ dom (𝐹 ∘ 𝐺) → 𝑋 ∈ dom 𝐺) |
9 | ndmfv 6878 | . . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺) → ((𝐹 ∘ 𝐺)‘𝑋) = ∅) | |
10 | 8, 9 | nsyl5 159 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → ((𝐹 ∘ 𝐺)‘𝑋) = ∅) |
11 | ndmfv 6878 | . . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺‘𝑋) = ∅) | |
12 | 11 | fveq2d 6847 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺‘𝑋)) = (𝐹‘∅)) |
13 | 6, 10, 12 | 3eqtr4a 2799 | . 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 1542 ∈ wcel 2107 ∅c0 4283 dom cdm 5634 ∘ ccom 5638 Fun wfun 6491 Fn wfn 6492 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 |
This theorem is referenced by: lidlval 20677 rspval 20678 |
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