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Mirrors > Home > ILE Home > Th. List > eloprabga | Unicode version |
Description: The law of concretion for operation class abstraction. Compare elopab 4013. (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 2610 | . 2 | |
2 | elex 2610 | . 2 | |
3 | elex 2610 | . 2 | |
4 | opexg 3983 | . . . . 5 | |
5 | opexg 3983 | . . . . 5 | |
6 | 4, 5 | sylan 277 | . . . 4 |
7 | 6 | 3impa 1133 | . . 3 |
8 | simpr 108 | . . . . . . . . . . 11 | |
9 | 8 | eqeq1d 2089 | . . . . . . . . . 10 |
10 | eqcom 2083 | . . . . . . . . . . 11 | |
11 | vex 2604 | . . . . . . . . . . . 12 | |
12 | vex 2604 | . . . . . . . . . . . 12 | |
13 | vex 2604 | . . . . . . . . . . . 12 | |
14 | 11, 12, 13 | otth2 3996 | . . . . . . . . . . 11 |
15 | 10, 14 | bitri 182 | . . . . . . . . . 10 |
16 | 9, 15 | syl6bb 194 | . . . . . . . . 9 |
17 | 16 | anbi1d 452 | . . . . . . . 8 |
18 | eloprabga.1 | . . . . . . . . 9 | |
19 | 18 | pm5.32i 441 | . . . . . . . 8 |
20 | 17, 19 | syl6bb 194 | . . . . . . 7 |
21 | 20 | 3exbidv 1790 | . . . . . 6 |
22 | df-oprab 5536 | . . . . . . . . . 10 | |
23 | 22 | eleq2i 2145 | . . . . . . . . 9 |
24 | abid 2069 | . . . . . . . . 9 | |
25 | 23, 24 | bitr2i 183 | . . . . . . . 8 |
26 | eleq1 2141 | . . . . . . . 8 | |
27 | 25, 26 | syl5bb 190 | . . . . . . 7 |
28 | 27 | adantl 271 | . . . . . 6 |
29 | elisset 2613 | . . . . . . . . . . 11 | |
30 | elisset 2613 | . . . . . . . . . . 11 | |
31 | elisset 2613 | . . . . . . . . . . 11 | |
32 | 29, 30, 31 | 3anim123i 1123 | . . . . . . . . . 10 |
33 | eeeanv 1849 | . . . . . . . . . 10 | |
34 | 32, 33 | sylibr 132 | . . . . . . . . 9 |
35 | 34 | biantrurd 299 | . . . . . . . 8 |
36 | 19.41vvv 1825 | . . . . . . . 8 | |
37 | 35, 36 | syl6rbbr 197 | . . . . . . 7 |
38 | 37 | adantr 270 | . . . . . 6 |
39 | 21, 28, 38 | 3bitr3d 216 | . . . . 5 |
40 | 39 | expcom 114 | . . . 4 |
41 | 40 | vtocleg 2669 | . . 3 |
42 | 7, 41 | mpcom 36 | . 2 |
43 | 1, 2, 3, 42 | syl3an 1211 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wex 1421 wcel 1433 cab 2067 cvv 2601 cop 3401 coprab 5533 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-oprab 5536 |
This theorem is referenced by: eloprabg 5612 ovigg 5641 |
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