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Mirrors > Home > MPE Home > Th. List > ovg | Structured version Visualization version 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 6653 | . . . . 5 | |
2 | ovg.5 | . . . . . 6 | |
3 | 2 | fveq1i 6192 | . . . . 5 |
4 | 1, 3 | eqtri 2644 | . . . 4 |
5 | 4 | eqeq1i 2627 | . . 3 |
6 | eqeq2 2633 | . . . . . . . . . 10 | |
7 | opeq2 4403 | . . . . . . . . . . 11 | |
8 | 7 | eleq1d 2686 | . . . . . . . . . 10 |
9 | 6, 8 | bibi12d 335 | . . . . . . . . 9 |
10 | 9 | imbi2d 330 | . . . . . . . 8 |
11 | ovg.4 | . . . . . . . . . . . 12 | |
12 | 11 | ex 450 | . . . . . . . . . . 11 |
13 | 12 | alrimivv 1856 | . . . . . . . . . 10 |
14 | fnoprabg 6761 | . . . . . . . . . 10 | |
15 | 13, 14 | syl 17 | . . . . . . . . 9 |
16 | eleq1 2689 | . . . . . . . . . . . 12 | |
17 | 16 | anbi1d 741 | . . . . . . . . . . 11 |
18 | eleq1 2689 | . . . . . . . . . . . 12 | |
19 | 18 | anbi2d 740 | . . . . . . . . . . 11 |
20 | 17, 19 | opelopabg 4993 | . . . . . . . . . 10 |
21 | 20 | ibir 257 | . . . . . . . . 9 |
22 | fnopfvb 6237 | . . . . . . . . 9 | |
23 | 15, 21, 22 | syl2an 494 | . . . . . . . 8 |
24 | 10, 23 | vtoclg 3266 | . . . . . . 7 |
25 | 24 | com12 32 | . . . . . 6 |
26 | 25 | exp32 631 | . . . . 5 |
27 | 26 | 3imp2 1282 | . . . 4 |
28 | ovg.1 | . . . . . . 7 | |
29 | 17, 28 | anbi12d 747 | . . . . . 6 |
30 | ovg.2 | . . . . . . 7 | |
31 | 19, 30 | anbi12d 747 | . . . . . 6 |
32 | ovg.3 | . . . . . . 7 | |
33 | 32 | anbi2d 740 | . . . . . 6 |
34 | 29, 31, 33 | eloprabg 6748 | . . . . 5 |
35 | 34 | adantl 482 | . . . 4 |
36 | 27, 35 | bitrd 268 | . . 3 |
37 | 5, 36 | syl5bb 272 | . 2 |
38 | biidd 252 | . . . . 5 | |
39 | 38 | bianabs 924 | . . . 4 |
40 | 39 | 3adant3 1081 | . . 3 |
41 | 40 | adantl 482 | . 2 |
42 | 37, 41 | bitrd 268 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wal 1481 wceq 1483 wcel 1990 weu 2470 cop 4183 copab 4712 wfn 5883 cfv 5888 (class class class)co 6650 coprab 6651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-ov 6653 df-oprab 6654 |
This theorem is referenced by: (None) |
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