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Mirrors > Home > ILE Home > Th. List > ovg | Unicode version |
Description: The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
ovg.1 | |
ovg.2 | |
ovg.3 | |
ovg.4 | |
ovg.5 |
Ref | Expression |
---|---|
ovg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 5515 | . . . . 5 | |
2 | ovg.5 | . . . . . 6 | |
3 | 2 | fveq1i 5179 | . . . . 5 |
4 | 1, 3 | eqtri 2060 | . . . 4 |
5 | 4 | eqeq1i 2047 | . . 3 |
6 | eqeq2 2049 | . . . . . . . . . 10 | |
7 | opeq2 3550 | . . . . . . . . . . 11 | |
8 | 7 | eleq1d 2106 | . . . . . . . . . 10 |
9 | 6, 8 | bibi12d 224 | . . . . . . . . 9 |
10 | 9 | imbi2d 219 | . . . . . . . 8 |
11 | ovg.4 | . . . . . . . . . . . 12 | |
12 | 11 | ex 108 | . . . . . . . . . . 11 |
13 | 12 | alrimivv 1755 | . . . . . . . . . 10 |
14 | fnoprabg 5602 | . . . . . . . . . 10 | |
15 | 13, 14 | syl 14 | . . . . . . . . 9 |
16 | eleq1 2100 | . . . . . . . . . . . 12 | |
17 | 16 | anbi1d 438 | . . . . . . . . . . 11 |
18 | eleq1 2100 | . . . . . . . . . . . 12 | |
19 | 18 | anbi2d 437 | . . . . . . . . . . 11 |
20 | 17, 19 | opelopabg 4005 | . . . . . . . . . 10 |
21 | 20 | ibir 166 | . . . . . . . . 9 |
22 | fnopfvb 5215 | . . . . . . . . 9 | |
23 | 15, 21, 22 | syl2an 273 | . . . . . . . 8 |
24 | 10, 23 | vtoclg 2613 | . . . . . . 7 |
25 | 24 | com12 27 | . . . . . 6 |
26 | 25 | exp32 347 | . . . . 5 |
27 | 26 | 3imp2 1119 | . . . 4 |
28 | ovg.1 | . . . . . . 7 | |
29 | 17, 28 | anbi12d 442 | . . . . . 6 |
30 | ovg.2 | . . . . . . 7 | |
31 | 19, 30 | anbi12d 442 | . . . . . 6 |
32 | ovg.3 | . . . . . . 7 | |
33 | 32 | anbi2d 437 | . . . . . 6 |
34 | 29, 31, 33 | eloprabg 5592 | . . . . 5 |
35 | 34 | adantl 262 | . . . 4 |
36 | 27, 35 | bitrd 177 | . . 3 |
37 | 5, 36 | syl5bb 181 | . 2 |
38 | biidd 161 | . . . . 5 | |
39 | 38 | bianabs 543 | . . . 4 |
40 | 39 | 3adant3 924 | . . 3 |
41 | 40 | adantl 262 | . 2 |
42 | 37, 41 | bitrd 177 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wal 1241 wceq 1243 wcel 1393 weu 1900 cop 3378 copab 3817 wfn 4897 cfv 4902 (class class class)co 5512 coprab 5513 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 df-ov 5515 df-oprab 5516 |
This theorem is referenced by: (None) |
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