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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 5777 | . . . . 5 | |
2 | ovg.5 | . . . . . 6 | |
3 | 2 | fveq1i 5422 | . . . . 5 |
4 | 1, 3 | eqtri 2160 | . . . 4 |
5 | 4 | eqeq1i 2147 | . . 3 |
6 | eqeq2 2149 | . . . . . . . . . 10 | |
7 | opeq2 3706 | . . . . . . . . . . 11 | |
8 | 7 | eleq1d 2208 | . . . . . . . . . 10 |
9 | 6, 8 | bibi12d 234 | . . . . . . . . 9 |
10 | 9 | imbi2d 229 | . . . . . . . 8 |
11 | ovg.4 | . . . . . . . . . . . 12 | |
12 | 11 | ex 114 | . . . . . . . . . . 11 |
13 | 12 | alrimivv 1847 | . . . . . . . . . 10 |
14 | fnoprabg 5872 | . . . . . . . . . 10 | |
15 | 13, 14 | syl 14 | . . . . . . . . 9 |
16 | eleq1 2202 | . . . . . . . . . . . 12 | |
17 | 16 | anbi1d 460 | . . . . . . . . . . 11 |
18 | eleq1 2202 | . . . . . . . . . . . 12 | |
19 | 18 | anbi2d 459 | . . . . . . . . . . 11 |
20 | 17, 19 | opelopabg 4190 | . . . . . . . . . 10 |
21 | 20 | ibir 176 | . . . . . . . . 9 |
22 | fnopfvb 5463 | . . . . . . . . 9 | |
23 | 15, 21, 22 | syl2an 287 | . . . . . . . 8 |
24 | 10, 23 | vtoclg 2746 | . . . . . . 7 |
25 | 24 | com12 30 | . . . . . 6 |
26 | 25 | exp32 362 | . . . . 5 |
27 | 26 | 3imp2 1200 | . . . 4 |
28 | ovg.1 | . . . . . . 7 | |
29 | 17, 28 | anbi12d 464 | . . . . . 6 |
30 | ovg.2 | . . . . . . 7 | |
31 | 19, 30 | anbi12d 464 | . . . . . 6 |
32 | ovg.3 | . . . . . . 7 | |
33 | 32 | anbi2d 459 | . . . . . 6 |
34 | 29, 31, 33 | eloprabg 5859 | . . . . 5 |
35 | 34 | adantl 275 | . . . 4 |
36 | 27, 35 | bitrd 187 | . . 3 |
37 | 5, 36 | syl5bb 191 | . 2 |
38 | biidd 171 | . . . . 5 | |
39 | 38 | bianabs 600 | . . . 4 |
40 | 39 | 3adant3 1001 | . . 3 |
41 | 40 | adantl 275 | . 2 |
42 | 37, 41 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wal 1329 wceq 1331 wcel 1480 weu 1999 cop 3530 copab 3988 wfn 5118 cfv 5123 (class class class)co 5774 coprab 5775 |
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-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 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-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 df-ov 5777 df-oprab 5778 |
This theorem is referenced by: (None) |
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