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Mirrors > Home > ILE Home > Th. List > caucvgprprlemval | Unicode version |
Description: Lemma for caucvgprpr 7520. 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 7132 | . . . . 5 | |
2 | 1 | brel 4591 | . . . 4 |
3 | 2 | adantl 275 | . . 3 |
4 | caucvgprpr.f | . . . . 5 | |
5 | caucvgprpr.cau | . . . . 5 | |
6 | 4, 5 | caucvgprprlemcbv 7495 | . . . 4 |
7 | 6 | adantr 274 | . . 3 |
8 | simpr 109 | . . 3 | |
9 | breq1 3932 | . . . . 5 | |
10 | fveq2 5421 | . . . . . . 7 | |
11 | opeq1 3705 | . . . . . . . . . . . . 13 | |
12 | 11 | eceq1d 6465 | . . . . . . . . . . . 12 |
13 | 12 | fveq2d 5425 | . . . . . . . . . . 11 |
14 | 13 | breq2d 3941 | . . . . . . . . . 10 |
15 | 14 | abbidv 2257 | . . . . . . . . 9 |
16 | 13 | breq1d 3939 | . . . . . . . . . 10 |
17 | 16 | abbidv 2257 | . . . . . . . . 9 |
18 | 15, 17 | opeq12d 3713 | . . . . . . . 8 |
19 | 18 | oveq2d 5790 | . . . . . . 7 |
20 | 10, 19 | breq12d 3942 | . . . . . 6 |
21 | 10, 18 | oveq12d 5792 | . . . . . . 7 |
22 | 21 | breq2d 3941 | . . . . . 6 |
23 | 20, 22 | anbi12d 464 | . . . . 5 |
24 | 9, 23 | imbi12d 233 | . . . 4 |
25 | breq2 3933 | . . . . 5 | |
26 | fveq2 5421 | . . . . . . . 8 | |
27 | 26 | oveq1d 5789 | . . . . . . 7 |
28 | 27 | breq2d 3941 | . . . . . 6 |
29 | 26 | breq1d 3939 | . . . . . 6 |
30 | 28, 29 | anbi12d 464 | . . . . 5 |
31 | 25, 30 | imbi12d 233 | . . . 4 |
32 | 24, 31 | rspc2v 2802 | . . 3 |
33 | 3, 7, 8, 32 | syl3c 63 | . 2 |
34 | breq1 3932 | . . . . . . 7 | |
35 | 34 | cbvabv 2264 | . . . . . 6 |
36 | breq2 3933 | . . . . . . 7 | |
37 | 36 | cbvabv 2264 | . . . . . 6 |
38 | 35, 37 | opeq12i 3710 | . . . . 5 |
39 | 38 | oveq2i 5785 | . . . 4 |
40 | 39 | breq2i 3937 | . . 3 |
41 | 38 | oveq2i 5785 | . . . 4 |
42 | 41 | breq2i 3937 | . . 3 |
43 | 40, 42 | anbi12i 455 | . 2 |
44 | 33, 43 | sylib 121 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cab 2125 wral 2416 cop 3530 class class class wbr 3929 wf 5119 cfv 5123 (class class class)co 5774 c1o 6306 cec 6427 cnpi 7080 clti 7083 ceq 7087 crq 7092 cltq 7093 cnp 7099 cpp 7101 cltp 7103 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fv 5131 df-ov 5777 df-ec 6431 df-lti 7115 |
This theorem is referenced by: caucvgprprlemnkltj 7497 caucvgprprlemnjltk 7499 caucvgprprlemnbj 7501 |
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