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Mirrors > Home > ILE Home > Th. List > caucvgprprlemval | Unicode version |
Description: Lemma for caucvgprpr 6902. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.) |
Ref | Expression |
---|---|
caucvgprpr.f | |
caucvgprpr.cau |
Ref | Expression |
---|---|
caucvgprprlemval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelpi 6514 | . . . . 5 | |
2 | 1 | brel 4410 | . . . 4 |
3 | 2 | adantl 271 | . . 3 |
4 | caucvgprpr.f | . . . . 5 | |
5 | caucvgprpr.cau | . . . . 5 | |
6 | 4, 5 | caucvgprprlemcbv 6877 | . . . 4 |
7 | 6 | adantr 270 | . . 3 |
8 | simpr 108 | . . 3 | |
9 | breq1 3788 | . . . . 5 | |
10 | fveq2 5198 | . . . . . . 7 | |
11 | opeq1 3570 | . . . . . . . . . . . . 13 | |
12 | 11 | eceq1d 6165 | . . . . . . . . . . . 12 |
13 | 12 | fveq2d 5202 | . . . . . . . . . . 11 |
14 | 13 | breq2d 3797 | . . . . . . . . . 10 |
15 | 14 | abbidv 2196 | . . . . . . . . 9 |
16 | 13 | breq1d 3795 | . . . . . . . . . 10 |
17 | 16 | abbidv 2196 | . . . . . . . . 9 |
18 | 15, 17 | opeq12d 3578 | . . . . . . . 8 |
19 | 18 | oveq2d 5548 | . . . . . . 7 |
20 | 10, 19 | breq12d 3798 | . . . . . 6 |
21 | 10, 18 | oveq12d 5550 | . . . . . . 7 |
22 | 21 | breq2d 3797 | . . . . . 6 |
23 | 20, 22 | anbi12d 456 | . . . . 5 |
24 | 9, 23 | imbi12d 232 | . . . 4 |
25 | breq2 3789 | . . . . 5 | |
26 | fveq2 5198 | . . . . . . . 8 | |
27 | 26 | oveq1d 5547 | . . . . . . 7 |
28 | 27 | breq2d 3797 | . . . . . 6 |
29 | 26 | breq1d 3795 | . . . . . 6 |
30 | 28, 29 | anbi12d 456 | . . . . 5 |
31 | 25, 30 | imbi12d 232 | . . . 4 |
32 | 24, 31 | rspc2v 2713 | . . 3 |
33 | 3, 7, 8, 32 | syl3c 62 | . 2 |
34 | breq1 3788 | . . . . . . 7 | |
35 | 34 | cbvabv 2202 | . . . . . 6 |
36 | breq2 3789 | . . . . . . 7 | |
37 | 36 | cbvabv 2202 | . . . . . 6 |
38 | 35, 37 | opeq12i 3575 | . . . . 5 |
39 | 38 | oveq2i 5543 | . . . 4 |
40 | 39 | breq2i 3793 | . . 3 |
41 | 38 | oveq2i 5543 | . . . 4 |
42 | 41 | breq2i 3793 | . . 3 |
43 | 40, 42 | anbi12i 447 | . 2 |
44 | 33, 43 | sylib 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cab 2067 wral 2348 cop 3401 class class class wbr 3785 wf 4918 cfv 4922 (class class class)co 5532 c1o 6017 cec 6127 cnpi 6462 clti 6465 ceq 6469 crq 6474 cltq 6475 cnp 6481 cpp 6483 cltp 6485 |
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-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 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 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-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fv 4930 df-ov 5535 df-ec 6131 df-lti 6497 |
This theorem is referenced by: caucvgprprlemnkltj 6879 caucvgprprlemnjltk 6881 caucvgprprlemnbj 6883 |
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