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Mirrors > Home > ILE Home > Th. List > erovlem | Unicode version |
Description: Lemma for eroprf 6199. (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 |
Ref | Expression |
---|---|
erovlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 102 | . . . . . . . 8 | |
2 | 1 | reximi 2416 | . . . . . . 7 |
3 | 2 | reximi 2416 | . . . . . 6 |
4 | eropr.1 | . . . . . . . . . 10 | |
5 | 4 | eleq2i 2104 | . . . . . . . . 9 |
6 | vex 2560 | . . . . . . . . . 10 | |
7 | 6 | elqs 6157 | . . . . . . . . 9 |
8 | 5, 7 | bitri 173 | . . . . . . . 8 |
9 | eropr.2 | . . . . . . . . . 10 | |
10 | 9 | eleq2i 2104 | . . . . . . . . 9 |
11 | vex 2560 | . . . . . . . . . 10 | |
12 | 11 | elqs 6157 | . . . . . . . . 9 |
13 | 10, 12 | bitri 173 | . . . . . . . 8 |
14 | 8, 13 | anbi12i 433 | . . . . . . 7 |
15 | reeanv 2479 | . . . . . . 7 | |
16 | 14, 15 | bitr4i 176 | . . . . . 6 |
17 | 3, 16 | sylibr 137 | . . . . 5 |
18 | 17 | pm4.71ri 372 | . . . 4 |
19 | eropr.3 | . . . . . . . 8 | |
20 | eropr.4 | . . . . . . . 8 | |
21 | eropr.5 | . . . . . . . 8 | |
22 | eropr.6 | . . . . . . . 8 | |
23 | eropr.7 | . . . . . . . 8 | |
24 | eropr.8 | . . . . . . . 8 | |
25 | eropr.9 | . . . . . . . 8 | |
26 | eropr.10 | . . . . . . . 8 | |
27 | eropr.11 | . . . . . . . 8 | |
28 | 4, 9, 19, 20, 21, 22, 23, 24, 25, 26, 27 | eroveu 6197 | . . . . . . 7 |
29 | iota1 4881 | . . . . . . 7 | |
30 | 28, 29 | syl 14 | . . . . . 6 |
31 | eqcom 2042 | . . . . . 6 | |
32 | 30, 31 | syl6bb 185 | . . . . 5 |
33 | 32 | pm5.32da 425 | . . . 4 |
34 | 18, 33 | syl5bb 181 | . . 3 |
35 | 34 | oprabbidv 5559 | . 2 |
36 | eropr.12 | . 2 | |
37 | df-mpt2 5517 | . . 3 | |
38 | nfv 1421 | . . . 4 | |
39 | nfv 1421 | . . . . 5 | |
40 | nfiota1 4869 | . . . . . 6 | |
41 | 40 | nfeq2 2189 | . . . . 5 |
42 | 39, 41 | nfan 1457 | . . . 4 |
43 | eqeq1 2046 | . . . . 5 | |
44 | 43 | anbi2d 437 | . . . 4 |
45 | 38, 42, 44 | cbvoprab3 5580 | . . 3 |
46 | 37, 45 | eqtr4i 2063 | . 2 |
47 | 35, 36, 46 | 3eqtr4g 2097 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 weu 1900 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-v 2559 df-sbc 2765 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-br 3765 df-opab 3819 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-er 6106 df-ec 6108 df-qs 6112 |
This theorem is referenced by: eroprf 6199 |
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