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Mirrors > Home > ILE Home > Th. List > caucvgprprlemval | Unicode version |
Description: Lemma for caucvgprpr 7513. 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 7125 | . . . . 5 | |
2 | 1 | brel 4586 | . . . 4 |
3 | 2 | adantl 275 | . . 3 |
4 | caucvgprpr.f | . . . . 5 | |
5 | caucvgprpr.cau | . . . . 5 | |
6 | 4, 5 | caucvgprprlemcbv 7488 | . . . 4 |
7 | 6 | adantr 274 | . . 3 |
8 | simpr 109 | . . 3 | |
9 | breq1 3927 | . . . . 5 | |
10 | fveq2 5414 | . . . . . . 7 | |
11 | opeq1 3700 | . . . . . . . . . . . . 13 | |
12 | 11 | eceq1d 6458 | . . . . . . . . . . . 12 |
13 | 12 | fveq2d 5418 | . . . . . . . . . . 11 |
14 | 13 | breq2d 3936 | . . . . . . . . . 10 |
15 | 14 | abbidv 2255 | . . . . . . . . 9 |
16 | 13 | breq1d 3934 | . . . . . . . . . 10 |
17 | 16 | abbidv 2255 | . . . . . . . . 9 |
18 | 15, 17 | opeq12d 3708 | . . . . . . . 8 |
19 | 18 | oveq2d 5783 | . . . . . . 7 |
20 | 10, 19 | breq12d 3937 | . . . . . 6 |
21 | 10, 18 | oveq12d 5785 | . . . . . . 7 |
22 | 21 | breq2d 3936 | . . . . . 6 |
23 | 20, 22 | anbi12d 464 | . . . . 5 |
24 | 9, 23 | imbi12d 233 | . . . 4 |
25 | breq2 3928 | . . . . 5 | |
26 | fveq2 5414 | . . . . . . . 8 | |
27 | 26 | oveq1d 5782 | . . . . . . 7 |
28 | 27 | breq2d 3936 | . . . . . 6 |
29 | 26 | breq1d 3934 | . . . . . 6 |
30 | 28, 29 | anbi12d 464 | . . . . 5 |
31 | 25, 30 | imbi12d 233 | . . . 4 |
32 | 24, 31 | rspc2v 2797 | . . 3 |
33 | 3, 7, 8, 32 | syl3c 63 | . 2 |
34 | breq1 3927 | . . . . . . 7 | |
35 | 34 | cbvabv 2262 | . . . . . 6 |
36 | breq2 3928 | . . . . . . 7 | |
37 | 36 | cbvabv 2262 | . . . . . 6 |
38 | 35, 37 | opeq12i 3705 | . . . . 5 |
39 | 38 | oveq2i 5778 | . . . 4 |
40 | 39 | breq2i 3932 | . . 3 |
41 | 38 | oveq2i 5778 | . . . 4 |
42 | 41 | breq2i 3932 | . . 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 2123 wral 2414 cop 3525 class class class wbr 3924 wf 5114 cfv 5118 (class class class)co 5767 c1o 6299 cec 6420 cnpi 7073 clti 7076 ceq 7080 crq 7085 cltq 7086 cnp 7092 cpp 7094 cltp 7096 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fv 5126 df-ov 5770 df-ec 6424 df-lti 7108 |
This theorem is referenced by: caucvgprprlemnkltj 7490 caucvgprprlemnjltk 7492 caucvgprprlemnbj 7494 |
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