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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 485 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
4 | inidm 4218 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
5 | eqidd 2733 | . . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝐹‘𝑋)) | |
6 | eqidd 2733 | . . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) = (𝐺‘𝑋)) | |
7 | 1, 2, 3, 3, 4, 5, 6 | ofval 7683 | . 2 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
8 | 7 | anasss 467 | 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 396 = wceq 1541 ∈ wcel 2106 Fn wfn 6538 ‘cfv 6543 (class class class)co 7411 ∘f cof 7670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 |
This theorem is referenced by: suppofssd 8190 ofccat 14920 ghmplusg 19755 lcomfsupp 20656 lmhmplusg 20799 frlmvplusgvalc 21541 frlmvscaval 21542 frlmsslsp 21570 frlmup1 21572 frlmup2 21573 islindf4 21612 evlslem3 21862 evlslem1 21864 coe1addfv 22007 evl1addd 22080 evl1subd 22081 evl1muld 22082 mamudi 22123 mamudir 22124 mdetrlin 22324 nmotri 24476 mdegaddle 25816 ply1rem 25905 fta1glem2 25908 fta1blem 25910 plyexmo 26050 ulmdvlem1 26136 jensen 26717 dchrmulcl 26976 dchrinv 26988 sumdchr2 26997 dchr2sum 27000 evlsaddval 41442 evlsmulval 41443 evladdval 41449 evlmulval 41450 mzpsubst 41788 mzpcong 42013 rngunsnply 42217 ofoafg 42406 ofoafo 42408 ofoaid1 42410 ofoaid2 42411 ofoaass 42412 ofoacom 42413 naddcnff 42414 naddcnffo 42416 naddcnfcom 42418 naddcnfid1 42419 naddcnfass 42421 lincsum 47198 |
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