Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5272 | . 2 ⊢ (𝐺:𝐴⟶𝐵 → 𝐺 Fn 𝐴) | |
2 | fvco2 5490 | . 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 1331 ∈ wcel 1480 ∘ ccom 4543 Fn wfn 5118 ⟶wf 5119 ‘cfv 5123 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 |
This theorem is referenced by: fvco4 5493 foco2 5655 f1ocnvfv1 5678 f1ocnvfv2 5679 fcof1 5684 fcofo 5685 cocan1 5688 cocan2 5689 isotr 5717 algrflem 6126 algrflemg 6127 difinfsn 6985 ctssdccl 6996 0tonninf 10212 1tonninf 10213 summodclem3 11149 fsumf1o 11159 fsumcl2lem 11167 fsumadd 11175 fsummulc2 11217 prodmodclem3 11344 algcvg 11729 ennnfonelemnn0 11935 ctinfomlemom 11940 cnptopco 12391 lmtopcnp 12419 upxp 12441 uptx 12443 cnmpt11 12452 cnmpt21 12460 comet 12668 cnmetdval 12698 climcncf 12740 cncfco 12747 limccnpcntop 12813 dvcoapbr 12840 dvcjbr 12841 dvfre 12843 isomninnlem 13225 |
Copyright terms: Public domain | W3C validator |