Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fco | Unicode version |
Description: Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fco |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 5186 | . . 3 | |
2 | df-f 5186 | . . 3 | |
3 | fnco 5290 | . . . . . . 7 | |
4 | 3 | 3expib 1195 | . . . . . 6 |
5 | 4 | adantr 274 | . . . . 5 |
6 | rncoss 4868 | . . . . . . 7 | |
7 | sstr 3145 | . . . . . . 7 | |
8 | 6, 7 | mpan 421 | . . . . . 6 |
9 | 8 | adantl 275 | . . . . 5 |
10 | 5, 9 | jctird 315 | . . . 4 |
11 | 10 | imp 123 | . . 3 |
12 | 1, 2, 11 | syl2anb 289 | . 2 |
13 | df-f 5186 | . 2 | |
14 | 12, 13 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wss 3111 crn 4599 ccom 4602 wfn 5177 wf 5178 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-fun 5184 df-fn 5185 df-f 5186 |
This theorem is referenced by: fco2 5348 f1co 5399 foco 5414 mapen 6803 ctm 7065 enomnilem 7093 enmkvlem 7116 enwomnilem 7124 fnn0nninf 10362 fsumcl2lem 11325 fsumadd 11333 fprodmul 11518 algcvg 11959 cnco 12768 cnptopco 12769 lmtopcnp 12797 cnmpt11 12830 cnmpt21 12838 comet 13046 cnmet 13077 cncfco 13125 limccnpcntop 13191 dvcoapbr 13218 dvcjbr 13219 dvcj 13220 |
Copyright terms: Public domain | W3C validator |