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Mirrors > Home > ILE Home > Th. List > eloprabga | Unicode version |
Description: The law of concretion for operation class abstraction. Compare elopab 3995. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
eloprabga.1 |
Ref | Expression |
---|---|
eloprabga |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2566 | . 2 | |
2 | elex 2566 | . 2 | |
3 | elex 2566 | . 2 | |
4 | opexg 3964 | . . . . 5 | |
5 | opexg 3964 | . . . . 5 | |
6 | 4, 5 | sylan 267 | . . . 4 |
7 | 6 | 3impa 1099 | . . 3 |
8 | simpr 103 | . . . . . . . . . . 11 | |
9 | 8 | eqeq1d 2048 | . . . . . . . . . 10 |
10 | eqcom 2042 | . . . . . . . . . . 11 | |
11 | vex 2560 | . . . . . . . . . . . 12 | |
12 | vex 2560 | . . . . . . . . . . . 12 | |
13 | vex 2560 | . . . . . . . . . . . 12 | |
14 | 11, 12, 13 | otth2 3978 | . . . . . . . . . . 11 |
15 | 10, 14 | bitri 173 | . . . . . . . . . 10 |
16 | 9, 15 | syl6bb 185 | . . . . . . . . 9 |
17 | 16 | anbi1d 438 | . . . . . . . 8 |
18 | eloprabga.1 | . . . . . . . . 9 | |
19 | 18 | pm5.32i 427 | . . . . . . . 8 |
20 | 17, 19 | syl6bb 185 | . . . . . . 7 |
21 | 20 | 3exbidv 1749 | . . . . . 6 |
22 | df-oprab 5516 | . . . . . . . . . 10 | |
23 | 22 | eleq2i 2104 | . . . . . . . . 9 |
24 | abid 2028 | . . . . . . . . 9 | |
25 | 23, 24 | bitr2i 174 | . . . . . . . 8 |
26 | eleq1 2100 | . . . . . . . 8 | |
27 | 25, 26 | syl5bb 181 | . . . . . . 7 |
28 | 27 | adantl 262 | . . . . . 6 |
29 | elisset 2568 | . . . . . . . . . . 11 | |
30 | elisset 2568 | . . . . . . . . . . 11 | |
31 | elisset 2568 | . . . . . . . . . . 11 | |
32 | 29, 30, 31 | 3anim123i 1089 | . . . . . . . . . 10 |
33 | eeeanv 1808 | . . . . . . . . . 10 | |
34 | 32, 33 | sylibr 137 | . . . . . . . . 9 |
35 | 34 | biantrurd 289 | . . . . . . . 8 |
36 | 19.41vvv 1784 | . . . . . . . 8 | |
37 | 35, 36 | syl6rbbr 188 | . . . . . . 7 |
38 | 37 | adantr 261 | . . . . . 6 |
39 | 21, 28, 38 | 3bitr3d 207 | . . . . 5 |
40 | 39 | expcom 109 | . . . 4 |
41 | 40 | vtocleg 2624 | . . 3 |
42 | 7, 41 | mpcom 32 | . 2 |
43 | 1, 2, 3, 42 | syl3an 1177 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wex 1381 wcel 1393 cab 2026 cvv 2557 cop 3378 coprab 5513 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-oprab 5516 |
This theorem is referenced by: eloprabg 5592 ovigg 5621 |
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