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Mirrors > Home > MPE Home > Th. List > fnfvof | Structured version Visualization version GIF version |
Description: Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.) |
Ref | Expression |
---|---|
fnfvof | ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 765 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐹 Fn 𝐴) | |
2 | simplr 767 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐺 Fn 𝐴) | |
3 | simpr 483 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
4 | inidm 4217 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
5 | eqidd 2726 | . . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝐹‘𝑋)) | |
6 | eqidd 2726 | . . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) = (𝐺‘𝑋)) | |
7 | 1, 2, 3, 3, 4, 5, 6 | ofval 7696 | . 2 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
8 | 7 | anasss 465 | 1 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Fn wfn 6544 ‘cfv 6549 (class class class)co 7419 ∘f cof 7683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 |
This theorem is referenced by: suppofssd 8209 ofccat 14952 ghmplusg 19813 lcomfsupp 20797 lmhmplusg 20941 frlmvplusgvalc 21718 frlmvscaval 21719 frlmsslsp 21747 frlmup1 21749 frlmup2 21750 islindf4 21789 evlslem3 22048 evlslem1 22050 coe1addfv 22209 evl1addd 22285 evl1subd 22286 evl1muld 22287 mamudi 22347 mamudir 22348 mdetrlin 22548 nmotri 24700 mdegaddle 26054 ply1rem 26145 fta1glem2 26148 fta1blem 26150 plyexmo 26293 ulmdvlem1 26381 jensen 26966 dchrmulcl 27227 dchrinv 27239 sumdchr2 27248 dchr2sum 27251 evlsaddval 41933 evlsmulval 41934 evladdval 41940 evlmulval 41941 mzpsubst 42307 mzpcong 42532 rngunsnply 42736 ofoafg 42922 ofoafo 42924 ofoaid1 42926 ofoaid2 42927 ofoaass 42928 ofoacom 42929 naddcnff 42930 naddcnffo 42932 naddcnfcom 42934 naddcnfid1 42935 naddcnfass 42937 lincsum 47680 |
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