Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fco | Structured version Visualization version GIF version |
Description: Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fco | ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 5808 | . . 3 ⊢ (𝐹:𝐵⟶𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶)) | |
2 | df-f 5808 | . . 3 ⊢ (𝐺:𝐴⟶𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵)) | |
3 | fnco 5913 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴) | |
4 | 3 | 3expib 1260 | . . . . . 6 ⊢ (𝐹 Fn 𝐵 → ((𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴)) |
5 | 4 | adantr 480 | . . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ((𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴)) |
6 | rncoss 5307 | . . . . . . 7 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 | |
7 | sstr 3576 | . . . . . . 7 ⊢ ((ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ 𝐶) → ran (𝐹 ∘ 𝐺) ⊆ 𝐶) | |
8 | 6, 7 | mpan 702 | . . . . . 6 ⊢ (ran 𝐹 ⊆ 𝐶 → ran (𝐹 ∘ 𝐺) ⊆ 𝐶) |
9 | 8 | adantl 481 | . . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ran (𝐹 ∘ 𝐺) ⊆ 𝐶) |
10 | 5, 9 | jctird 565 | . . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ((𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶))) |
11 | 10 | imp 444 | . . 3 ⊢ (((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) ∧ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵)) → ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶)) |
12 | 1, 2, 11 | syl2anb 495 | . 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶)) |
13 | df-f 5808 | . 2 ⊢ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ↔ ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶)) | |
14 | 12, 13 | sylibr 223 | 1 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ⊆ wss 3540 ran crn 5039 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 |
This theorem is referenced by: fco2 5972 f1co 6023 foco 6038 mapen 8009 fsuppco2 8191 mapfienlem1 8193 mapfienlem3 8195 mapfien 8196 unxpwdom2 8376 wemapwe 8477 cofsmo 8974 cfcoflem 8977 isf34lem7 9084 isf34lem6 9085 canthp1lem2 9354 inar1 9476 addnqf 9649 mulnqf 9650 axdc4uzlem 12644 seqf1olem2 12703 wrdco 13428 lenco 13429 lo1o1 14111 o1co 14165 caucvgrlem2 14253 fsumcl2lem 14309 fsumadd 14317 fsummulc2 14358 fsumrelem 14380 supcvg 14427 fprodcl2lem 14519 fprodmul 14529 fproddiv 14530 fprodn0 14548 algcvg 15127 cofucl 16371 setccatid 16557 estrccatid 16595 funcestrcsetclem9 16611 funcsetcestrclem9 16626 yonedalem3b 16742 mhmco 17185 pwsco1mhm 17193 pwsco2mhm 17194 gsumwmhm 17205 f1omvdconj 17689 pmtrfinv 17704 symgtrinv 17715 psgnunilem1 17736 gsumval3lem1 18129 gsumval3lem2 18130 gsumval3 18131 gsumzcl2 18134 gsumzf1o 18136 gsumzaddlem 18144 gsumzmhm 18160 gsumzoppg 18167 gsumzinv 18168 gsumsub 18171 dprdf1o 18254 ablfaclem2 18308 psrass1lem 19198 psrnegcl 19217 coe1f2 19400 cnfldds 19577 dsmmbas2 19900 f1lindf 19980 lindfmm 19985 cpmadumatpolylem1 20505 cnco 20880 cnpco 20881 lmcnp 20918 cnmpt11 21276 cnmpt21 21284 qtopcn 21327 fmco 21575 flfcnp 21618 tsmsf1o 21758 tsmsmhm 21759 tsmssub 21762 imasdsf1olem 21988 comet 22128 nrmmetd 22189 isngp2 22211 isngp3 22212 tngngp2 22266 cnmet 22385 cnfldms 22389 cncfco 22518 cnfldcusp 22961 ovolfioo 23043 ovolficc 23044 ovolfsf 23047 ovollb 23054 ovolctb 23065 ovolicc2lem4 23095 ovolicc2 23097 volsup 23131 uniioovol 23153 uniioombllem3a 23158 uniioombllem3 23159 uniioombllem4 23160 uniioombllem5 23161 uniioombl 23163 mbfdm 23201 ismbfcn 23204 mbfres 23217 mbfimaopnlem 23228 cncombf 23231 limccnp 23461 dvcobr 23515 dvcof 23517 dvcjbr 23518 dvcj 23519 dvmptco 23541 dvlip2 23562 itgsubstlem 23615 coecj 23838 pserulm 23980 jensenlem2 24514 jensen 24515 amgmlem 24516 gamf 24569 dchrinv 24786 motcgrg 25239 vsfval 26872 imsdf 26928 lnocoi 26996 hocofi 28009 homco1 28044 homco2 28220 hmopco 28266 kbass2 28360 kbass5 28363 opsqrlem1 28383 opsqrlem6 28388 pjinvari 28434 fcobij 28888 fcobijfs 28889 mbfmco 29653 dstfrvclim1 29866 subfacp1lem5 30420 mrsubco 30672 mclsppslem 30734 circum 30822 mblfinlem2 32617 mbfresfi 32626 ftc1anclem5 32659 ghomco 32860 rngohomco 32943 tendococl 35078 mapco2g 36295 diophrw 36340 hausgraph 36809 sblpnf 37531 fcoss 38397 fcod 38419 limccog 38687 mbfres2cn 38850 volioof 38880 volioofmpt 38887 voliooicof 38889 stoweidlem31 38924 stoweidlem59 38952 subsaliuncllem 39251 sge0resrnlem 39296 hoicvr 39438 ovolval2lem 39533 ovolval2 39534 ovolval3 39537 ovolval4lem1 39539 ovolval5lem3 39544 mgmhmco 41591 amgmwlem 42357 |
Copyright terms: Public domain | W3C validator |