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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 6597 | . 2 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺) | |
2 | fvco2 7005 | . 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 1536 ∈ wcel 2105 dom cdm 5688 ∘ ccom 5692 Fun wfun 6556 Fn wfn 6557 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-fv 6570 |
This theorem is referenced by: fin23lem30 10379 hashkf 14367 hashgval 14368 gsumpropd2lem 18704 ofco2 22472 opfv 32660 xppreima 32661 psgnfzto1stlem 33102 cycpmfv1 33115 cycpmfv2 33116 cyc3co2 33142 smatlem 33757 mdetpmtr1 33783 madjusmdetlem2 33788 madjusmdetlem4 33790 eulerpartlemgvv 34357 eulerpartlemgu 34358 sseqfv2 34375 reprpmtf1o 34619 hgt750lemg 34647 aks5lem2 42168 comptiunov2i 43695 choicefi 45142 fvcod 45169 evthiccabs 45448 cncficcgt0 45843 dvsinax 45868 fvvolioof 45944 fvvolicof 45946 stirlinglem14 46042 fourierdlem42 46104 hoicvr 46503 hoi2toco 46562 ovolval3 46602 ovolval4lem1 46604 ovnovollem1 46611 ovnovollem2 46612 |
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