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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 5535 | . . . . 5 | |
2 | ovg.5 | . . . . . 6 | |
3 | 2 | fveq1i 5199 | . . . . 5 |
4 | 1, 3 | eqtri 2101 | . . . 4 |
5 | 4 | eqeq1i 2088 | . . 3 |
6 | eqeq2 2090 | . . . . . . . . . 10 | |
7 | opeq2 3571 | . . . . . . . . . . 11 | |
8 | 7 | eleq1d 2147 | . . . . . . . . . 10 |
9 | 6, 8 | bibi12d 233 | . . . . . . . . 9 |
10 | 9 | imbi2d 228 | . . . . . . . 8 |
11 | ovg.4 | . . . . . . . . . . . 12 | |
12 | 11 | ex 113 | . . . . . . . . . . 11 |
13 | 12 | alrimivv 1796 | . . . . . . . . . 10 |
14 | fnoprabg 5622 | . . . . . . . . . 10 | |
15 | 13, 14 | syl 14 | . . . . . . . . 9 |
16 | eleq1 2141 | . . . . . . . . . . . 12 | |
17 | 16 | anbi1d 452 | . . . . . . . . . . 11 |
18 | eleq1 2141 | . . . . . . . . . . . 12 | |
19 | 18 | anbi2d 451 | . . . . . . . . . . 11 |
20 | 17, 19 | opelopabg 4023 | . . . . . . . . . 10 |
21 | 20 | ibir 175 | . . . . . . . . 9 |
22 | fnopfvb 5236 | . . . . . . . . 9 | |
23 | 15, 21, 22 | syl2an 283 | . . . . . . . 8 |
24 | 10, 23 | vtoclg 2658 | . . . . . . 7 |
25 | 24 | com12 30 | . . . . . 6 |
26 | 25 | exp32 357 | . . . . 5 |
27 | 26 | 3imp2 1153 | . . . 4 |
28 | ovg.1 | . . . . . . 7 | |
29 | 17, 28 | anbi12d 456 | . . . . . 6 |
30 | ovg.2 | . . . . . . 7 | |
31 | 19, 30 | anbi12d 456 | . . . . . 6 |
32 | ovg.3 | . . . . . . 7 | |
33 | 32 | anbi2d 451 | . . . . . 6 |
34 | 29, 31, 33 | eloprabg 5612 | . . . . 5 |
35 | 34 | adantl 271 | . . . 4 |
36 | 27, 35 | bitrd 186 | . . 3 |
37 | 5, 36 | syl5bb 190 | . 2 |
38 | biidd 170 | . . . . 5 | |
39 | 38 | bianabs 575 | . . . 4 |
40 | 39 | 3adant3 958 | . . 3 |
41 | 40 | adantl 271 | . 2 |
42 | 37, 41 | bitrd 186 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wal 1282 wceq 1284 wcel 1433 weu 1941 cop 3401 copab 3838 wfn 4917 cfv 4922 (class class class)co 5532 coprab 5533 |
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-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 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-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 df-ov 5535 df-oprab 5536 |
This theorem is referenced by: (None) |
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