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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 457 | . . . . . . . . . . 11 |
3 | eropr.10 | . . . . . . . . . . . . 13 | |
4 | 3 | adantr 261 | . . . . . . . . . . . 12 |
5 | 4 | fovrnda 5644 | . . . . . . . . . . 11 |
6 | ecelqsg 6159 | . . . . . . . . . . 11 | |
7 | 2, 5, 6 | syl2anc 391 | . . . . . . . . . 10 |
8 | eropr.15 | . . . . . . . . . 10 | |
9 | 7, 8 | syl6eleqr 2131 | . . . . . . . . 9 |
10 | eleq1a 2109 | . . . . . . . . 9 | |
11 | 9, 10 | syl 14 | . . . . . . . 8 |
12 | 11 | adantld 263 | . . . . . . 7 |
13 | 12 | rexlimdvva 2440 | . . . . . 6 |
14 | 13 | abssdv 3014 | . . . . 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 6197 | . . . . . 6 |
25 | iotacl 4890 | . . . . . 6 | |
26 | 24, 25 | syl 14 | . . . . 5 |
27 | 14, 26 | sseldd 2946 | . . . 4 |
28 | 27 | ralrimivva 2401 | . . 3 |
29 | eqid 2040 | . . . 4 | |
30 | 29 | fmpt2 5827 | . . 3 |
31 | 28, 30 | sylib 127 | . 2 |
32 | eropr.12 | . . . 4 | |
33 | 15, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23, 32 | erovlem 6198 | . . 3 |
34 | 33 | feq1d 5034 | . 2 |
35 | 31, 34 | mpbird 156 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 weu 1900 cab 2026 wral 2306 wrex 2307 wss 2917 class class class wbr 3764 cxp 4343 cio 4865 wf 4898 (class class class)co 5512 coprab 5513 cmpt2 5514 wer 6103 cec 6104 cqs 6105 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-er 6106 df-ec 6108 df-qs 6112 |
This theorem is referenced by: eroprf2 6200 |
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