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Mirrors > Home > ILE Home > Th. List > fmpoco | Unicode version |
Description: Composition of two functions. Variation of fmptco 5586 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.) |
Ref | Expression |
---|---|
fmpoco.1 | |
fmpoco.2 | |
fmpoco.3 | |
fmpoco.4 |
Ref | Expression |
---|---|
fmpoco |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmpoco.1 | . . . . . 6 | |
2 | 1 | ralrimivva 2514 | . . . . 5 |
3 | eqid 2139 | . . . . . 6 | |
4 | 3 | fmpo 6099 | . . . . 5 |
5 | 2, 4 | sylib 121 | . . . 4 |
6 | nfcv 2281 | . . . . . . 7 | |
7 | nfcv 2281 | . . . . . . 7 | |
8 | nfcv 2281 | . . . . . . . 8 | |
9 | nfcsb1v 3035 | . . . . . . . 8 | |
10 | 8, 9 | nfcsb 3037 | . . . . . . 7 |
11 | nfcsb1v 3035 | . . . . . . 7 | |
12 | csbeq1a 3012 | . . . . . . . 8 | |
13 | csbeq1a 3012 | . . . . . . . 8 | |
14 | 12, 13 | sylan9eq 2192 | . . . . . . 7 |
15 | 6, 7, 10, 11, 14 | cbvmpo 5850 | . . . . . 6 |
16 | vex 2689 | . . . . . . . . . 10 | |
17 | vex 2689 | . . . . . . . . . 10 | |
18 | 16, 17 | op2ndd 6047 | . . . . . . . . 9 |
19 | 18 | csbeq1d 3010 | . . . . . . . 8 |
20 | 16, 17 | op1std 6046 | . . . . . . . . . 10 |
21 | 20 | csbeq1d 3010 | . . . . . . . . 9 |
22 | 21 | csbeq2dv 3028 | . . . . . . . 8 |
23 | 19, 22 | eqtrd 2172 | . . . . . . 7 |
24 | 23 | mpompt 5863 | . . . . . 6 |
25 | 15, 24 | eqtr4i 2163 | . . . . 5 |
26 | 25 | fmpt 5570 | . . . 4 |
27 | 5, 26 | sylibr 133 | . . 3 |
28 | fmpoco.2 | . . . 4 | |
29 | 28, 25 | syl6eq 2188 | . . 3 |
30 | fmpoco.3 | . . 3 | |
31 | 27, 29, 30 | fmptcos 5588 | . 2 |
32 | 23 | csbeq1d 3010 | . . . . 5 |
33 | 32 | mpompt 5863 | . . . 4 |
34 | nfcv 2281 | . . . . 5 | |
35 | nfcv 2281 | . . . . 5 | |
36 | nfcv 2281 | . . . . . 6 | |
37 | 10, 36 | nfcsb 3037 | . . . . 5 |
38 | nfcv 2281 | . . . . . 6 | |
39 | 11, 38 | nfcsb 3037 | . . . . 5 |
40 | 14 | csbeq1d 3010 | . . . . 5 |
41 | 34, 35, 37, 39, 40 | cbvmpo 5850 | . . . 4 |
42 | 33, 41 | eqtr4i 2163 | . . 3 |
43 | 1 | 3impb 1177 | . . . . 5 |
44 | nfcvd 2282 | . . . . . 6 | |
45 | fmpoco.4 | . . . . . 6 | |
46 | 44, 45 | csbiegf 3043 | . . . . 5 |
47 | 43, 46 | syl 14 | . . . 4 |
48 | 47 | mpoeq3dva 5835 | . . 3 |
49 | 42, 48 | syl5eq 2184 | . 2 |
50 | 31, 49 | eqtrd 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wral 2416 csb 3003 cop 3530 cmpt 3989 cxp 4537 ccom 4543 wf 5119 cfv 5123 cmpo 5776 c1st 6036 c2nd 6037 |
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-13 1491 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 ax-un 4355 |
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-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 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-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 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 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: oprabco 6114 txswaphmeolem 12489 bdxmet 12670 |
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