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Mirrors > Home > ILE Home > Th. List > fvco3 | GIF version |
Description: Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.) |
Ref | Expression |
---|---|
fvco3 | ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5319 | . 2 ⊢ (𝐺:𝐴⟶𝐵 → 𝐺 Fn 𝐴) | |
2 | fvco2 5537 | . 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) | |
3 | 1, 2 | sylan 281 | 1 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 ∈ wcel 2128 ∘ ccom 4590 Fn wfn 5165 ⟶wf 5166 ‘cfv 5170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fv 5178 |
This theorem is referenced by: fvco4 5540 foco2 5704 f1ocnvfv1 5727 f1ocnvfv2 5728 fcof1 5733 fcofo 5734 cocan1 5737 cocan2 5738 isotr 5766 algrflem 6176 algrflemg 6177 difinfsn 7044 ctssdccl 7055 cc3 7188 0tonninf 10338 1tonninf 10339 summodclem3 11277 fsumf1o 11287 fsumcl2lem 11295 fsumadd 11303 fsummulc2 11345 prodmodclem3 11472 fprodf1o 11485 fprodmul 11488 algcvg 11925 eulerthlemth 12107 ennnfonelemnn0 12162 ctinfomlemom 12167 cnptopco 12633 lmtopcnp 12661 upxp 12683 uptx 12685 cnmpt11 12694 cnmpt21 12702 comet 12910 cnmetdval 12940 climcncf 12982 cncfco 12989 limccnpcntop 13055 dvcoapbr 13082 dvcjbr 13083 dvfre 13085 isomninnlem 13612 iswomninnlem 13631 ismkvnnlem 13634 |
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