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Mirrors > Home > MPE Home > Th. List > resfunexg | Structured version Visualization version Unicode version |
Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.) |
Ref | Expression |
---|---|
resfunexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funres 5929 | . . . . . . 7 | |
2 | 1 | adantr 481 | . . . . . 6 |
3 | funfn 5918 | . . . . . 6 | |
4 | 2, 3 | sylib 208 | . . . . 5 |
5 | dffn5 6241 | . . . . 5 | |
6 | 4, 5 | sylib 208 | . . . 4 |
7 | fvex 6201 | . . . . 5 | |
8 | 7 | fnasrn 6411 | . . . 4 |
9 | 6, 8 | syl6eq 2672 | . . 3 |
10 | opex 4932 | . . . . . 6 | |
11 | eqid 2622 | . . . . . 6 | |
12 | 10, 11 | dmmpti 6023 | . . . . 5 |
13 | 12 | imaeq2i 5464 | . . . 4 |
14 | imadmrn 5476 | . . . 4 | |
15 | 13, 14 | eqtr3i 2646 | . . 3 |
16 | 9, 15 | syl6eqr 2674 | . 2 |
17 | funmpt 5926 | . . 3 | |
18 | dmresexg 5421 | . . . 4 | |
19 | 18 | adantl 482 | . . 3 |
20 | funimaexg 5975 | . . 3 | |
21 | 17, 19, 20 | sylancr 695 | . 2 |
22 | 16, 21 | eqeltrd 2701 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 cop 4183 cmpt 4729 cdm 5114 crn 5115 cres 5116 cima 5117 wfun 5882 wfn 5883 cfv 5888 |
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-rep 4771 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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 |
This theorem is referenced by: resiexd 6480 fnex 6481 ofexg 6901 cofunexg 7130 dfac8alem 8852 dfac12lem1 8965 cfsmolem 9092 alephsing 9098 itunifval 9238 zorn2lem1 9318 ttukeylem3 9333 imadomg 9356 wunex2 9560 inar1 9597 axdc4uzlem 12782 hashf1rn 13142 hashf1rnOLD 13143 bpolylem 14779 1stf1 16832 1stf2 16833 2ndf1 16835 2ndf2 16836 1stfcl 16837 2ndfcl 16838 gsumzadd 18322 wlkreslem 26566 madeval 31935 tendo02 36075 dnnumch1 37614 aomclem6 37629 dfrngc2 41972 dfringc2 42018 rngcresringcat 42030 fdivval 42333 |
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