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Mirrors > Home > MPE Home > Th. List > mptpreima | Structured version Visualization version Unicode version |
Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
dmmpt.1 |
Ref | Expression |
---|---|
mptpreima |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmmpt.1 | . . . . . 6 | |
2 | df-mpt 4730 | . . . . . 6 | |
3 | 1, 2 | eqtri 2644 | . . . . 5 |
4 | 3 | cnveqi 5297 | . . . 4 |
5 | cnvopab 5533 | . . . 4 | |
6 | 4, 5 | eqtri 2644 | . . 3 |
7 | 6 | imaeq1i 5463 | . 2 |
8 | df-ima 5127 | . . 3 | |
9 | resopab 5446 | . . . . 5 | |
10 | 9 | rneqi 5352 | . . . 4 |
11 | ancom 466 | . . . . . . . . 9 | |
12 | anass 681 | . . . . . . . . 9 | |
13 | 11, 12 | bitri 264 | . . . . . . . 8 |
14 | 13 | exbii 1774 | . . . . . . 7 |
15 | 19.42v 1918 | . . . . . . . 8 | |
16 | df-clel 2618 | . . . . . . . . . 10 | |
17 | 16 | bicomi 214 | . . . . . . . . 9 |
18 | 17 | anbi2i 730 | . . . . . . . 8 |
19 | 15, 18 | bitri 264 | . . . . . . 7 |
20 | 14, 19 | bitri 264 | . . . . . 6 |
21 | 20 | abbii 2739 | . . . . 5 |
22 | rnopab 5370 | . . . . 5 | |
23 | df-rab 2921 | . . . . 5 | |
24 | 21, 22, 23 | 3eqtr4i 2654 | . . . 4 |
25 | 10, 24 | eqtri 2644 | . . 3 |
26 | 8, 25 | eqtri 2644 | . 2 |
27 | 7, 26 | eqtri 2644 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wceq 1483 wex 1704 wcel 1990 cab 2608 crab 2916 copab 4712 cmpt 4729 ccnv 5113 crn 5115 cres 5116 cima 5117 |
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-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 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-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 |
This theorem is referenced by: mptiniseg 5629 dmmpt 5630 fmpt 6381 f1oresrab 6395 mptsuppdifd 7317 r0weon 8835 compss 9198 infrenegsup 11006 eqglact 17645 odngen 17992 psrbagsn 19495 coe1mul2lem2 19638 pjdm 20051 xkoccn 21422 txcnmpt 21427 txdis1cn 21438 pthaus 21441 txkgen 21455 xkoco1cn 21460 xkoco2cn 21461 xkoinjcn 21490 txconn 21492 imasnopn 21493 imasncld 21494 imasncls 21495 ptcmplem1 21856 ptcmplem3 21858 ptcmplem4 21859 tmdgsum2 21900 symgtgp 21905 tgpconncompeqg 21915 ghmcnp 21918 tgpt0 21922 qustgpopn 21923 qustgphaus 21926 eltsms 21936 prdsxmslem2 22334 efopn 24404 atansopn 24659 xrlimcnp 24695 fpwrelmapffslem 29507 ptrest 33408 mbfposadd 33457 cnambfre 33458 itg2addnclem2 33462 iblabsnclem 33473 ftc1anclem1 33485 ftc1anclem6 33490 pwfi2f1o 37666 smfpimioo 40994 |
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