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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 766 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐹 Fn 𝐴) | |
2 | simplr 768 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐺 Fn 𝐴) | |
3 | simpr 486 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
4 | inidm 4219 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
5 | eqidd 2734 | . . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝐹‘𝑋)) | |
6 | eqidd 2734 | . . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) = (𝐺‘𝑋)) | |
7 | 1, 2, 3, 3, 4, 5, 6 | ofval 7681 | . 2 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
8 | 7 | anasss 468 | 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 397 = wceq 1542 ∈ wcel 2107 Fn wfn 6539 ‘cfv 6544 (class class class)co 7409 ∘f cof 7668 |
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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 |
This theorem is referenced by: suppofssd 8188 ofccat 14916 ghmplusg 19714 lcomfsupp 20512 lmhmplusg 20655 frlmvplusgvalc 21322 frlmvscaval 21323 frlmsslsp 21351 frlmup1 21353 frlmup2 21354 islindf4 21393 evlslem3 21643 evlslem1 21645 coe1addfv 21787 evl1addd 21860 evl1subd 21861 evl1muld 21862 mamudi 21903 mamudir 21904 mdetrlin 22104 nmotri 24256 mdegaddle 25592 ply1rem 25681 fta1glem2 25684 fta1blem 25686 plyexmo 25826 ulmdvlem1 25912 jensen 26493 dchrmulcl 26752 dchrinv 26764 sumdchr2 26773 dchr2sum 26776 evlsaddval 41140 evlsmulval 41141 evladdval 41147 evlmulval 41148 mzpsubst 41486 mzpcong 41711 rngunsnply 41915 ofoafg 42104 ofoafo 42106 ofoaid1 42108 ofoaid2 42109 ofoaass 42110 ofoacom 42111 naddcnff 42112 naddcnffo 42114 naddcnfcom 42116 naddcnfid1 42117 naddcnfass 42119 lincsum 47110 |
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