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Mirrors > Home > ILE Home > Th. List > eloprabi | Unicode version |
Description: A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
eloprabi.1 | |
eloprabi.2 | |
eloprabi.3 |
Ref | Expression |
---|---|
eloprabi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2146 | . . . . . 6 | |
2 | 1 | anbi1d 460 | . . . . 5 |
3 | 2 | 3exbidv 1841 | . . . 4 |
4 | df-oprab 5778 | . . . 4 | |
5 | 3, 4 | elab2g 2831 | . . 3 |
6 | 5 | ibi 175 | . 2 |
7 | vex 2689 | . . . . . . . . . . . 12 | |
8 | vex 2689 | . . . . . . . . . . . 12 | |
9 | 7, 8 | opex 4151 | . . . . . . . . . . 11 |
10 | vex 2689 | . . . . . . . . . . 11 | |
11 | 9, 10 | op1std 6046 | . . . . . . . . . 10 |
12 | 11 | fveq2d 5425 | . . . . . . . . 9 |
13 | 7, 8 | op1st 6044 | . . . . . . . . 9 |
14 | 12, 13 | syl6req 2189 | . . . . . . . 8 |
15 | eloprabi.1 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | 11 | fveq2d 5425 | . . . . . . . . 9 |
18 | 7, 8 | op2nd 6045 | . . . . . . . . 9 |
19 | 17, 18 | syl6req 2189 | . . . . . . . 8 |
20 | eloprabi.2 | . . . . . . . 8 | |
21 | 19, 20 | syl 14 | . . . . . . 7 |
22 | 9, 10 | op2ndd 6047 | . . . . . . . . 9 |
23 | 22 | eqcomd 2145 | . . . . . . . 8 |
24 | eloprabi.3 | . . . . . . . 8 | |
25 | 23, 24 | syl 14 | . . . . . . 7 |
26 | 16, 21, 25 | 3bitrd 213 | . . . . . 6 |
27 | 26 | biimpa 294 | . . . . 5 |
28 | 27 | exlimiv 1577 | . . . 4 |
29 | 28 | exlimiv 1577 | . . 3 |
30 | 29 | exlimiv 1577 | . 2 |
31 | 6, 30 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cop 3530 cfv 5123 coprab 5775 c1st 6036 c2nd 6037 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-oprab 5778 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: (None) |
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