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Mirrors > Home > ILE Home > Th. List > eloprabga | Unicode version |
Description: The law of concretion for operation class abstraction. Compare elopab 4230. (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 2732 | . 2 | |
2 | elex 2732 | . 2 | |
3 | elex 2732 | . 2 | |
4 | opexg 4200 | . . . . 5 | |
5 | opexg 4200 | . . . . 5 | |
6 | 4, 5 | sylan 281 | . . . 4 |
7 | 6 | 3impa 1183 | . . 3 |
8 | simpr 109 | . . . . . . . . . . 11 | |
9 | 8 | eqeq1d 2173 | . . . . . . . . . 10 |
10 | eqcom 2166 | . . . . . . . . . . 11 | |
11 | vex 2724 | . . . . . . . . . . . 12 | |
12 | vex 2724 | . . . . . . . . . . . 12 | |
13 | vex 2724 | . . . . . . . . . . . 12 | |
14 | 11, 12, 13 | otth2 4213 | . . . . . . . . . . 11 |
15 | 10, 14 | bitri 183 | . . . . . . . . . 10 |
16 | 9, 15 | bitrdi 195 | . . . . . . . . 9 |
17 | 16 | anbi1d 461 | . . . . . . . 8 |
18 | eloprabga.1 | . . . . . . . . 9 | |
19 | 18 | pm5.32i 450 | . . . . . . . 8 |
20 | 17, 19 | bitrdi 195 | . . . . . . 7 |
21 | 20 | 3exbidv 1856 | . . . . . 6 |
22 | df-oprab 5840 | . . . . . . . . . 10 | |
23 | 22 | eleq2i 2231 | . . . . . . . . 9 |
24 | abid 2152 | . . . . . . . . 9 | |
25 | 23, 24 | bitr2i 184 | . . . . . . . 8 |
26 | eleq1 2227 | . . . . . . . 8 | |
27 | 25, 26 | syl5bb 191 | . . . . . . 7 |
28 | 27 | adantl 275 | . . . . . 6 |
29 | 19.41vvv 1891 | . . . . . . . 8 | |
30 | elisset 2735 | . . . . . . . . . . 11 | |
31 | elisset 2735 | . . . . . . . . . . 11 | |
32 | elisset 2735 | . . . . . . . . . . 11 | |
33 | 30, 31, 32 | 3anim123i 1173 | . . . . . . . . . 10 |
34 | eeeanv 1920 | . . . . . . . . . 10 | |
35 | 33, 34 | sylibr 133 | . . . . . . . . 9 |
36 | 35 | biantrurd 303 | . . . . . . . 8 |
37 | 29, 36 | bitr4id 198 | . . . . . . 7 |
38 | 37 | adantr 274 | . . . . . 6 |
39 | 21, 28, 38 | 3bitr3d 217 | . . . . 5 |
40 | 39 | expcom 115 | . . . 4 |
41 | 40 | vtocleg 2792 | . . 3 |
42 | 7, 41 | mpcom 36 | . 2 |
43 | 1, 2, 3, 42 | syl3an 1269 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wex 1479 wcel 2135 cab 2150 cvv 2721 cop 3573 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-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-oprab 5840 |
This theorem is referenced by: eloprabg 5921 ovigg 5953 |
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