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| Mirrors > Home > MPE Home > Th. List > fvco | Structured version Visualization version GIF version | ||
| Description: Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.) |
| Ref | Expression |
|---|---|
| fvco | ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝐴) = (𝐹‘(𝐺‘𝐴))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funfn 6596 | . 2 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺) | |
| 2 | fvco2 7006 | . 2 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝐴) = (𝐹‘(𝐺‘𝐴))) | |
| 3 | 1, 2 | sylanb 581 | 1 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝐴) = (𝐹‘(𝐺‘𝐴))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 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: fin23lem30 10382 hashkf 14371 hashgval 14372 gsumpropd2lem 18692 ofco2 22457 opfv 32654 xppreima 32655 psgnfzto1stlem 33120 cycpmfv1 33133 cycpmfv2 33134 cyc3co2 33160 smatlem 33796 mdetpmtr1 33822 madjusmdetlem2 33827 madjusmdetlem4 33829 eulerpartlemgvv 34378 eulerpartlemgu 34379 sseqfv2 34396 reprpmtf1o 34641 hgt750lemg 34669 aks5lem2 42188 comptiunov2i 43719 choicefi 45205 fvcod 45232 evthiccabs 45509 cncficcgt0 45903 dvsinax 45928 fvvolioof 46004 fvvolicof 46006 stirlinglem14 46102 fourierdlem42 46164 hoicvr 46563 hoi2toco 46622 ovolval3 46662 ovolval4lem1 46664 ovnovollem1 46671 ovnovollem2 46672 |
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