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Mirrors > Home > ILE Home > Th. List > erovlem | Unicode version |
Description: Lemma for eroprf 6530. (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 108 | . . . . . . . 8 | |
2 | 1 | reximi 2532 | . . . . . . 7 |
3 | 2 | reximi 2532 | . . . . . 6 |
4 | eropr.1 | . . . . . . . . . 10 | |
5 | 4 | eleq2i 2207 | . . . . . . . . 9 |
6 | vex 2692 | . . . . . . . . . 10 | |
7 | 6 | elqs 6488 | . . . . . . . . 9 |
8 | 5, 7 | bitri 183 | . . . . . . . 8 |
9 | eropr.2 | . . . . . . . . . 10 | |
10 | 9 | eleq2i 2207 | . . . . . . . . 9 |
11 | vex 2692 | . . . . . . . . . 10 | |
12 | 11 | elqs 6488 | . . . . . . . . 9 |
13 | 10, 12 | bitri 183 | . . . . . . . 8 |
14 | 8, 13 | anbi12i 456 | . . . . . . 7 |
15 | reeanv 2603 | . . . . . . 7 | |
16 | 14, 15 | bitr4i 186 | . . . . . 6 |
17 | 3, 16 | sylibr 133 | . . . . 5 |
18 | 17 | pm4.71ri 390 | . . . 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 6528 | . . . . . . 7 |
29 | iota1 5110 | . . . . . . 7 | |
30 | 28, 29 | syl 14 | . . . . . 6 |
31 | eqcom 2142 | . . . . . 6 | |
32 | 30, 31 | syl6bb 195 | . . . . 5 |
33 | 32 | pm5.32da 448 | . . . 4 |
34 | 18, 33 | syl5bb 191 | . . 3 |
35 | 34 | oprabbidv 5833 | . 2 |
36 | eropr.12 | . 2 | |
37 | df-mpo 5787 | . . 3 | |
38 | nfv 1509 | . . . 4 | |
39 | nfv 1509 | . . . . 5 | |
40 | nfiota1 5098 | . . . . . 6 | |
41 | 40 | nfeq2 2294 | . . . . 5 |
42 | 39, 41 | nfan 1545 | . . . 4 |
43 | eqeq1 2147 | . . . . 5 | |
44 | 43 | anbi2d 460 | . . . 4 |
45 | 38, 42, 44 | cbvoprab3 5855 | . . 3 |
46 | 37, 45 | eqtr4i 2164 | . 2 |
47 | 35, 36, 46 | 3eqtr4g 2198 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1332 wcel 1481 weu 2000 wrex 2418 wss 3076 class class class wbr 3937 cxp 4545 cio 5094 wf 5127 (class class class)co 5782 coprab 5783 cmpo 5784 wer 6434 cec 6435 cqs 6436 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-er 6437 df-ec 6439 df-qs 6443 |
This theorem is referenced by: eroprf 6530 |
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