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Mirrors > Home > MPE Home > Th. List > resoprab | Structured version Visualization version Unicode version |
Description: Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.) |
Ref | Expression |
---|---|
resoprab |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resopab 5446 | . . 3 | |
2 | 19.42vv 1920 | . . . . 5 | |
3 | an12 838 | . . . . . . 7 | |
4 | eleq1 2689 | . . . . . . . . . 10 | |
5 | opelxp 5146 | . . . . . . . . . 10 | |
6 | 4, 5 | syl6bb 276 | . . . . . . . . 9 |
7 | 6 | anbi1d 741 | . . . . . . . 8 |
8 | 7 | pm5.32i 669 | . . . . . . 7 |
9 | 3, 8 | bitri 264 | . . . . . 6 |
10 | 9 | 2exbii 1775 | . . . . 5 |
11 | 2, 10 | bitr3i 266 | . . . 4 |
12 | 11 | opabbii 4717 | . . 3 |
13 | 1, 12 | eqtri 2644 | . 2 |
14 | dfoprab2 6701 | . . 3 | |
15 | 14 | reseq1i 5392 | . 2 |
16 | dfoprab2 6701 | . 2 | |
17 | 13, 15, 16 | 3eqtr4i 2654 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wceq 1483 wex 1704 wcel 1990 cop 4183 copab 4712 cxp 5112 cres 5116 coprab 6651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-xp 5120 df-rel 5121 df-res 5126 df-oprab 6654 |
This theorem is referenced by: resoprab2 6757 df1stres 29481 df2ndres 29482 |
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