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 5347 | . 2 ⊢ (𝐺:𝐴⟶𝐵 → 𝐺 Fn 𝐴) | |
2 | fvco2 5565 | . 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 1348 ∈ wcel 2141 ∘ ccom 4615 Fn wfn 5193 ⟶wf 5194 ‘cfv 5198 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 |
This theorem is referenced by: fvco4 5568 foco2 5733 f1ocnvfv1 5756 f1ocnvfv2 5757 fcof1 5762 fcofo 5763 cocan1 5766 cocan2 5767 isotr 5795 algrflem 6208 algrflemg 6209 difinfsn 7077 ctssdccl 7088 cc3 7230 0tonninf 10395 1tonninf 10396 summodclem3 11343 fsumf1o 11353 fsumcl2lem 11361 fsumadd 11369 fsummulc2 11411 prodmodclem3 11538 fprodf1o 11551 fprodmul 11554 algcvg 12002 eulerthlemth 12186 ennnfonelemnn0 12377 ctinfomlemom 12382 mhmco 12705 cnptopco 13016 lmtopcnp 13044 upxp 13066 uptx 13068 cnmpt11 13077 cnmpt21 13085 comet 13293 cnmetdval 13323 climcncf 13365 cncfco 13372 limccnpcntop 13438 dvcoapbr 13465 dvcjbr 13466 dvfre 13468 isomninnlem 14062 iswomninnlem 14081 ismkvnnlem 14084 |
Copyright terms: Public domain | W3C validator |