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Mirrors > Home > MPE Home > Th. List > f1imacnv | Structured version Visualization version Unicode version |
Description: Preimage of an image. (Contributed by NM, 30-Sep-2004.) |
Ref | Expression |
---|---|
f1imacnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resima 5431 | . 2 | |
2 | df-f1 5893 | . . . . . . 7 | |
3 | 2 | simprbi 480 | . . . . . 6 |
4 | 3 | adantr 481 | . . . . 5 |
5 | funcnvres 5967 | . . . . 5 | |
6 | 4, 5 | syl 17 | . . . 4 |
7 | 6 | imaeq1d 5465 | . . 3 |
8 | f1ores 6151 | . . . . 5 | |
9 | f1ocnv 6149 | . . . . 5 | |
10 | 8, 9 | syl 17 | . . . 4 |
11 | imadmrn 5476 | . . . . 5 | |
12 | f1odm 6141 | . . . . . 6 | |
13 | 12 | imaeq2d 5466 | . . . . 5 |
14 | f1ofo 6144 | . . . . . 6 | |
15 | forn 6118 | . . . . . 6 | |
16 | 14, 15 | syl 17 | . . . . 5 |
17 | 11, 13, 16 | 3eqtr3a 2680 | . . . 4 |
18 | 10, 17 | syl 17 | . . 3 |
19 | 7, 18 | eqtr3d 2658 | . 2 |
20 | 1, 19 | syl5eqr 2670 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wss 3574 ccnv 5113 cdm 5114 crn 5115 cres 5116 cima 5117 wfun 5882 wf 5884 wf1 5885 wfo 5886 wf1o 5887 |
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-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-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 |
This theorem is referenced by: f1opw2 6888 ssenen 8134 f1opwfi 8270 isf34lem3 9197 subggim 17708 gicsubgen 17721 cnt1 21154 basqtop 21514 tgqtop 21515 hmeoopn 21569 hmeocld 21570 hmeontr 21572 qtopf1 21619 f1otrg 25751 tpr2rico 29958 eulerpartlemmf 30437 ballotlemscr 30580 ballotlemrinv0 30594 cvmlift2lem9a 31285 grpokerinj 33692 |
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