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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 2177 | . . . . . 6 | |
2 | 1 | anbi1d 462 | . . . . 5 |
3 | 2 | 3exbidv 1862 | . . . 4 |
4 | df-oprab 5855 | . . . 4 | |
5 | 3, 4 | elab2g 2877 | . . 3 |
6 | 5 | ibi 175 | . 2 |
7 | vex 2733 | . . . . . . . . . . . 12 | |
8 | vex 2733 | . . . . . . . . . . . 12 | |
9 | 7, 8 | opex 4212 | . . . . . . . . . . 11 |
10 | vex 2733 | . . . . . . . . . . 11 | |
11 | 9, 10 | op1std 6125 | . . . . . . . . . 10 |
12 | 11 | fveq2d 5498 | . . . . . . . . 9 |
13 | 7, 8 | op1st 6123 | . . . . . . . . 9 |
14 | 12, 13 | eqtr2di 2220 | . . . . . . . 8 |
15 | eloprabi.1 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | 11 | fveq2d 5498 | . . . . . . . . 9 |
18 | 7, 8 | op2nd 6124 | . . . . . . . . 9 |
19 | 17, 18 | eqtr2di 2220 | . . . . . . . 8 |
20 | eloprabi.2 | . . . . . . . 8 | |
21 | 19, 20 | syl 14 | . . . . . . 7 |
22 | 9, 10 | op2ndd 6126 | . . . . . . . . 9 |
23 | 22 | eqcomd 2176 | . . . . . . . 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 1591 | . . . 4 |
29 | 28 | exlimiv 1591 | . . 3 |
30 | 29 | exlimiv 1591 | . 2 |
31 | 6, 30 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 cop 3584 cfv 5196 coprab 5852 c1st 6115 c2nd 6116 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-iota 5158 df-fun 5198 df-fv 5204 df-oprab 5855 df-1st 6117 df-2nd 6118 |
This theorem is referenced by: (None) |
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