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Mirrors > Home > ILE Home > Th. List > erovlem | Unicode version |
Description: Lemma for eroprf 6222. (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 107 | . . . . . . . 8 | |
2 | 1 | reximi 2458 | . . . . . . 7 |
3 | 2 | reximi 2458 | . . . . . 6 |
4 | eropr.1 | . . . . . . . . . 10 | |
5 | 4 | eleq2i 2145 | . . . . . . . . 9 |
6 | vex 2604 | . . . . . . . . . 10 | |
7 | 6 | elqs 6180 | . . . . . . . . 9 |
8 | 5, 7 | bitri 182 | . . . . . . . 8 |
9 | eropr.2 | . . . . . . . . . 10 | |
10 | 9 | eleq2i 2145 | . . . . . . . . 9 |
11 | vex 2604 | . . . . . . . . . 10 | |
12 | 11 | elqs 6180 | . . . . . . . . 9 |
13 | 10, 12 | bitri 182 | . . . . . . . 8 |
14 | 8, 13 | anbi12i 447 | . . . . . . 7 |
15 | reeanv 2523 | . . . . . . 7 | |
16 | 14, 15 | bitr4i 185 | . . . . . 6 |
17 | 3, 16 | sylibr 132 | . . . . 5 |
18 | 17 | pm4.71ri 384 | . . . 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 6220 | . . . . . . 7 |
29 | iota1 4901 | . . . . . . 7 | |
30 | 28, 29 | syl 14 | . . . . . 6 |
31 | eqcom 2083 | . . . . . 6 | |
32 | 30, 31 | syl6bb 194 | . . . . 5 |
33 | 32 | pm5.32da 439 | . . . 4 |
34 | 18, 33 | syl5bb 190 | . . 3 |
35 | 34 | oprabbidv 5579 | . 2 |
36 | eropr.12 | . 2 | |
37 | df-mpt2 5537 | . . 3 | |
38 | nfv 1461 | . . . 4 | |
39 | nfv 1461 | . . . . 5 | |
40 | nfiota1 4889 | . . . . . 6 | |
41 | 40 | nfeq2 2230 | . . . . 5 |
42 | 39, 41 | nfan 1497 | . . . 4 |
43 | eqeq1 2087 | . . . . 5 | |
44 | 43 | anbi2d 451 | . . . 4 |
45 | 38, 42, 44 | cbvoprab3 5600 | . . 3 |
46 | 37, 45 | eqtr4i 2104 | . 2 |
47 | 35, 36, 46 | 3eqtr4g 2138 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 weu 1941 wrex 2349 wss 2973 class class class wbr 3785 cxp 4361 cio 4885 wf 4918 (class class class)co 5532 coprab 5533 cmpt2 5534 wer 6126 cec 6127 cqs 6128 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-er 6129 df-ec 6131 df-qs 6135 |
This theorem is referenced by: eroprf 6222 |
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