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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 5868 | . . . . 5 | |
2 | ovg.5 | . . . . . 6 | |
3 | 2 | fveq1i 5508 | . . . . 5 |
4 | 1, 3 | eqtri 2196 | . . . 4 |
5 | 4 | eqeq1i 2183 | . . 3 |
6 | eqeq2 2185 | . . . . . . . . . 10 | |
7 | opeq2 3775 | . . . . . . . . . . 11 | |
8 | 7 | eleq1d 2244 | . . . . . . . . . 10 |
9 | 6, 8 | bibi12d 235 | . . . . . . . . 9 |
10 | 9 | imbi2d 230 | . . . . . . . 8 |
11 | ovg.4 | . . . . . . . . . . . 12 | |
12 | 11 | ex 115 | . . . . . . . . . . 11 |
13 | 12 | alrimivv 1873 | . . . . . . . . . 10 |
14 | fnoprabg 5966 | . . . . . . . . . 10 | |
15 | 13, 14 | syl 14 | . . . . . . . . 9 |
16 | eleq1 2238 | . . . . . . . . . . . 12 | |
17 | 16 | anbi1d 465 | . . . . . . . . . . 11 |
18 | eleq1 2238 | . . . . . . . . . . . 12 | |
19 | 18 | anbi2d 464 | . . . . . . . . . . 11 |
20 | 17, 19 | opelopabg 4262 | . . . . . . . . . 10 |
21 | 20 | ibir 177 | . . . . . . . . 9 |
22 | fnopfvb 5549 | . . . . . . . . 9 | |
23 | 15, 21, 22 | syl2an 289 | . . . . . . . 8 |
24 | 10, 23 | vtoclg 2795 | . . . . . . 7 |
25 | 24 | com12 30 | . . . . . 6 |
26 | 25 | exp32 365 | . . . . 5 |
27 | 26 | 3imp2 1222 | . . . 4 |
28 | ovg.1 | . . . . . . 7 | |
29 | 17, 28 | anbi12d 473 | . . . . . 6 |
30 | ovg.2 | . . . . . . 7 | |
31 | 19, 30 | anbi12d 473 | . . . . . 6 |
32 | ovg.3 | . . . . . . 7 | |
33 | 32 | anbi2d 464 | . . . . . 6 |
34 | 29, 31, 33 | eloprabg 5953 | . . . . 5 |
35 | 34 | adantl 277 | . . . 4 |
36 | 27, 35 | bitrd 188 | . . 3 |
37 | 5, 36 | bitrid 192 | . 2 |
38 | biidd 172 | . . . . 5 | |
39 | 38 | bianabs 611 | . . . 4 |
40 | 39 | 3adant3 1017 | . . 3 |
41 | 40 | adantl 277 | . 2 |
42 | 37, 41 | bitrd 188 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 w3a 978 wal 1351 wceq 1353 weu 2024 wcel 2146 cop 3592 copab 4058 wfn 5203 cfv 5208 (class class class)co 5865 coprab 5866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-sbc 2961 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-iota 5170 df-fun 5210 df-fn 5211 df-fv 5216 df-ov 5868 df-oprab 5869 |
This theorem is referenced by: (None) |
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