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Mirrors > Home > MPE Home > Th. List > foimacnv | Structured version Visualization version Unicode version |
Description: A reverse version of f1imacnv 6153. (Contributed by Jeff Hankins, 16-Jul-2009.) |
Ref | Expression |
---|---|
foimacnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resima 5431 | . 2 | |
2 | fofun 6116 | . . . . . 6 | |
3 | 2 | adantr 481 | . . . . 5 |
4 | funcnvres2 5969 | . . . . 5 | |
5 | 3, 4 | syl 17 | . . . 4 |
6 | 5 | imaeq1d 5465 | . . 3 |
7 | resss 5422 | . . . . . . . . . . 11 | |
8 | cnvss 5294 | . . . . . . . . . . 11 | |
9 | 7, 8 | ax-mp 5 | . . . . . . . . . 10 |
10 | cnvcnvss 5589 | . . . . . . . . . 10 | |
11 | 9, 10 | sstri 3612 | . . . . . . . . 9 |
12 | funss 5907 | . . . . . . . . 9 | |
13 | 11, 2, 12 | mpsyl 68 | . . . . . . . 8 |
14 | 13 | adantr 481 | . . . . . . 7 |
15 | df-ima 5127 | . . . . . . . 8 | |
16 | df-rn 5125 | . . . . . . . 8 | |
17 | 15, 16 | eqtr2i 2645 | . . . . . . 7 |
18 | 14, 17 | jctir 561 | . . . . . 6 |
19 | df-fn 5891 | . . . . . 6 | |
20 | 18, 19 | sylibr 224 | . . . . 5 |
21 | dfdm4 5316 | . . . . . 6 | |
22 | forn 6118 | . . . . . . . . . 10 | |
23 | 22 | sseq2d 3633 | . . . . . . . . 9 |
24 | 23 | biimpar 502 | . . . . . . . 8 |
25 | df-rn 5125 | . . . . . . . 8 | |
26 | 24, 25 | syl6sseq 3651 | . . . . . . 7 |
27 | ssdmres 5420 | . . . . . . 7 | |
28 | 26, 27 | sylib 208 | . . . . . 6 |
29 | 21, 28 | syl5eqr 2670 | . . . . 5 |
30 | df-fo 5894 | . . . . 5 | |
31 | 20, 29, 30 | sylanbrc 698 | . . . 4 |
32 | foima 6120 | . . . 4 | |
33 | 31, 32 | syl 17 | . . 3 |
34 | 6, 33 | eqtr3d 2658 | . 2 |
35 | 1, 34 | 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 wfn 5883 wfo 5886 |
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-fo 5894 |
This theorem is referenced by: f1opw2 6888 imacosupp 7335 fopwdom 8068 f1opwfi 8270 enfin2i 9143 fin1a2lem7 9228 fsumss 14456 fprodss 14678 gicsubgen 17721 coe1mul2lem2 19638 cncmp 21195 cnconn 21225 qtoprest 21520 qtopomap 21521 qtopcmap 21522 hmeoimaf1o 21573 elfm3 21754 imasf1oxms 22294 mbfimaopnlem 23422 cvmsss2 31256 diaintclN 36347 dibintclN 36456 dihintcl 36633 lnmepi 37655 pwfi2f1o 37666 sge0f1o 40599 |
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