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Mirrors > Home > ILE Home > Th. List > fmptcof | Unicode version |
Description: Version of fmptco 5586 where needn't be distinct from . (Contributed by NM, 27-Dec-2014.) |
Ref | Expression |
---|---|
fmptcof.1 | |
fmptcof.2 | |
fmptcof.3 | |
fmptcof.4 |
Ref | Expression |
---|---|
fmptcof |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptcof.1 | . . . . 5 | |
2 | nfcsb1v 3035 | . . . . . . 7 | |
3 | 2 | nfel1 2292 | . . . . . 6 |
4 | csbeq1a 3012 | . . . . . . 7 | |
5 | 4 | eleq1d 2208 | . . . . . 6 |
6 | 3, 5 | rspc 2783 | . . . . 5 |
7 | 1, 6 | mpan9 279 | . . . 4 |
8 | fmptcof.2 | . . . . 5 | |
9 | nfcv 2281 | . . . . . 6 | |
10 | 9, 2, 4 | cbvmpt 4023 | . . . . 5 |
11 | 8, 10 | syl6eq 2188 | . . . 4 |
12 | fmptcof.3 | . . . . 5 | |
13 | nfcv 2281 | . . . . . 6 | |
14 | nfcsb1v 3035 | . . . . . 6 | |
15 | csbeq1a 3012 | . . . . . 6 | |
16 | 13, 14, 15 | cbvmpt 4023 | . . . . 5 |
17 | 12, 16 | syl6eq 2188 | . . . 4 |
18 | csbeq1 3006 | . . . 4 | |
19 | 7, 11, 17, 18 | fmptco 5586 | . . 3 |
20 | nfcv 2281 | . . . 4 | |
21 | nfcv 2281 | . . . . 5 | |
22 | 2, 21 | nfcsb 3037 | . . . 4 |
23 | 4 | csbeq1d 3010 | . . . 4 |
24 | 20, 22, 23 | cbvmpt 4023 | . . 3 |
25 | 19, 24 | syl6eqr 2190 | . 2 |
26 | eqid 2139 | . . . 4 | |
27 | nfcvd 2282 | . . . . . 6 | |
28 | fmptcof.4 | . . . . . 6 | |
29 | 27, 28 | csbiegf 3043 | . . . . 5 |
30 | 29 | ralimi 2495 | . . . 4 |
31 | mpteq12 4011 | . . . 4 | |
32 | 26, 30, 31 | sylancr 410 | . . 3 |
33 | 1, 32 | syl 14 | . 2 |
34 | 25, 33 | eqtrd 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 wral 2416 csb 3003 cmpt 3989 ccom 4543 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
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-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 |
This theorem is referenced by: fmptcos 5588 cncfmpt1f 12753 sincn 12858 coscn 12859 |
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