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Mirrors > Home > ILE Home > Th. List > eroprf | Unicode version |
Description: Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
eropr.1 | |
eropr.2 | |
eropr.3 | |
eropr.4 | |
eropr.5 | |
eropr.6 | |
eropr.7 | |
eropr.8 | |
eropr.9 | |
eropr.10 | |
eropr.11 | |
eropr.12 | |
eropr.13 | |
eropr.14 | |
eropr.15 |
Ref | Expression |
---|---|
eroprf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eropr.3 | . . . . . . . . . . . 12 | |
2 | 1 | ad2antrr 479 | . . . . . . . . . . 11 |
3 | eropr.10 | . . . . . . . . . . . . 13 | |
4 | 3 | adantr 274 | . . . . . . . . . . . 12 |
5 | 4 | fovrnda 5914 | . . . . . . . . . . 11 |
6 | ecelqsg 6482 | . . . . . . . . . . 11 | |
7 | 2, 5, 6 | syl2anc 408 | . . . . . . . . . 10 |
8 | eropr.15 | . . . . . . . . . 10 | |
9 | 7, 8 | eleqtrrdi 2233 | . . . . . . . . 9 |
10 | eleq1a 2211 | . . . . . . . . 9 | |
11 | 9, 10 | syl 14 | . . . . . . . 8 |
12 | 11 | adantld 276 | . . . . . . 7 |
13 | 12 | rexlimdvva 2557 | . . . . . 6 |
14 | 13 | abssdv 3171 | . . . . 5 |
15 | eropr.1 | . . . . . . 7 | |
16 | eropr.2 | . . . . . . 7 | |
17 | eropr.4 | . . . . . . 7 | |
18 | eropr.5 | . . . . . . 7 | |
19 | eropr.6 | . . . . . . 7 | |
20 | eropr.7 | . . . . . . 7 | |
21 | eropr.8 | . . . . . . 7 | |
22 | eropr.9 | . . . . . . 7 | |
23 | eropr.11 | . . . . . . 7 | |
24 | 15, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23 | eroveu 6520 | . . . . . 6 |
25 | iotacl 5111 | . . . . . 6 | |
26 | 24, 25 | syl 14 | . . . . 5 |
27 | 14, 26 | sseldd 3098 | . . . 4 |
28 | 27 | ralrimivva 2514 | . . 3 |
29 | eqid 2139 | . . . 4 | |
30 | 29 | fmpo 6099 | . . 3 |
31 | 28, 30 | sylib 121 | . 2 |
32 | eropr.12 | . . . 4 | |
33 | 15, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23, 32 | erovlem 6521 | . . 3 |
34 | 33 | feq1d 5259 | . 2 |
35 | 31, 34 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 weu 1999 cab 2125 wral 2416 wrex 2417 wss 3071 class class class wbr 3929 cxp 4537 cio 5086 wf 5119 (class class class)co 5774 coprab 5775 cmpo 5776 wer 6426 cec 6427 cqs 6428 |
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-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-er 6429 df-ec 6431 df-qs 6435 |
This theorem is referenced by: eroprf2 6523 |
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