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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 5907 | . . . . . . . . . . 11 |
6 | ecelqsg 6475 | . . . . . . . . . . 11 | |
7 | 2, 5, 6 | syl2anc 408 | . . . . . . . . . 10 |
8 | eropr.15 | . . . . . . . . . 10 | |
9 | 7, 8 | eleqtrrdi 2231 | . . . . . . . . 9 |
10 | eleq1a 2209 | . . . . . . . . 9 | |
11 | 9, 10 | syl 14 | . . . . . . . 8 |
12 | 11 | adantld 276 | . . . . . . 7 |
13 | 12 | rexlimdvva 2555 | . . . . . 6 |
14 | 13 | abssdv 3166 | . . . . 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 6513 | . . . . . 6 |
25 | iotacl 5106 | . . . . . 6 | |
26 | 24, 25 | syl 14 | . . . . 5 |
27 | 14, 26 | sseldd 3093 | . . . 4 |
28 | 27 | ralrimivva 2512 | . . 3 |
29 | eqid 2137 | . . . 4 | |
30 | 29 | fmpo 6092 | . . 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 6514 | . . 3 |
34 | 33 | feq1d 5254 | . 2 |
35 | 31, 34 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 weu 1997 cab 2123 wral 2414 wrex 2415 wss 3066 class class class wbr 3924 cxp 4532 cio 5081 wf 5114 (class class class)co 5767 coprab 5768 cmpo 5769 wer 6419 cec 6420 cqs 6421 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
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-iun 3810 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 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-er 6422 df-ec 6424 df-qs 6428 |
This theorem is referenced by: eroprf2 6516 |
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