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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 5839 | . . . . 5 | |
2 | ovg.5 | . . . . . 6 | |
3 | 2 | fveq1i 5481 | . . . . 5 |
4 | 1, 3 | eqtri 2185 | . . . 4 |
5 | 4 | eqeq1i 2172 | . . 3 |
6 | eqeq2 2174 | . . . . . . . . . 10 | |
7 | opeq2 3753 | . . . . . . . . . . 11 | |
8 | 7 | eleq1d 2233 | . . . . . . . . . 10 |
9 | 6, 8 | bibi12d 234 | . . . . . . . . 9 |
10 | 9 | imbi2d 229 | . . . . . . . 8 |
11 | ovg.4 | . . . . . . . . . . . 12 | |
12 | 11 | ex 114 | . . . . . . . . . . 11 |
13 | 12 | alrimivv 1862 | . . . . . . . . . 10 |
14 | fnoprabg 5934 | . . . . . . . . . 10 | |
15 | 13, 14 | syl 14 | . . . . . . . . 9 |
16 | eleq1 2227 | . . . . . . . . . . . 12 | |
17 | 16 | anbi1d 461 | . . . . . . . . . . 11 |
18 | eleq1 2227 | . . . . . . . . . . . 12 | |
19 | 18 | anbi2d 460 | . . . . . . . . . . 11 |
20 | 17, 19 | opelopabg 4240 | . . . . . . . . . 10 |
21 | 20 | ibir 176 | . . . . . . . . 9 |
22 | fnopfvb 5522 | . . . . . . . . 9 | |
23 | 15, 21, 22 | syl2an 287 | . . . . . . . 8 |
24 | 10, 23 | vtoclg 2781 | . . . . . . 7 |
25 | 24 | com12 30 | . . . . . 6 |
26 | 25 | exp32 363 | . . . . 5 |
27 | 26 | 3imp2 1211 | . . . 4 |
28 | ovg.1 | . . . . . . 7 | |
29 | 17, 28 | anbi12d 465 | . . . . . 6 |
30 | ovg.2 | . . . . . . 7 | |
31 | 19, 30 | anbi12d 465 | . . . . . 6 |
32 | ovg.3 | . . . . . . 7 | |
33 | 32 | anbi2d 460 | . . . . . 6 |
34 | 29, 31, 33 | eloprabg 5921 | . . . . 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 601 | . . . 4 |
40 | 39 | 3adant3 1006 | . . 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 967 wal 1340 wceq 1342 weu 2013 wcel 2135 cop 3573 copab 4036 wfn 5177 cfv 5182 (class class class)co 5836 coprab 5837 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 df-ov 5839 df-oprab 5840 |
This theorem is referenced by: (None) |
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