Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fmptcos | Unicode version |
Description: Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fmptcof.1 | |
fmptcof.2 | |
fmptcof.3 |
Ref | Expression |
---|---|
fmptcos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptcof.1 | . 2 | |
2 | fmptcof.2 | . 2 | |
3 | fmptcof.3 | . . 3 | |
4 | nfcv 2279 | . . . 4 | |
5 | nfcsb1v 3030 | . . . 4 | |
6 | csbeq1a 3007 | . . . 4 | |
7 | 4, 5, 6 | cbvmpt 4018 | . . 3 |
8 | 3, 7 | syl6eq 2186 | . 2 |
9 | csbeq1 3001 | . 2 | |
10 | 1, 2, 8, 9 | fmptcof 5580 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 wral 2414 csb 2998 cmpt 3984 ccom 4538 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 |
This theorem is referenced by: fmpoco 6106 |
Copyright terms: Public domain | W3C validator |