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Mirrors > Home > NFE Home > Th. List > funssres | Unicode version |
Description: The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.) |
Ref | Expression |
---|---|
funssres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3268 | . . . . . 6 | |
2 | 1 | adantl 452 | . . . . 5 |
3 | opeldm 4911 | . . . . . 6 | |
4 | 3 | a1i 10 | . . . . 5 |
5 | 2, 4 | jcad 519 | . . . 4 |
6 | funeu2 5133 | . . . . . . . . . . 11 | |
7 | eldm2 4900 | . . . . . . . . . . . . 13 | |
8 | 1 | ancrd 537 | . . . . . . . . . . . . . 14 |
9 | 8 | eximdv 1622 | . . . . . . . . . . . . 13 |
10 | 7, 9 | syl5bi 208 | . . . . . . . . . . . 12 |
11 | 10 | imp 418 | . . . . . . . . . . 11 |
12 | eupick 2267 | . . . . . . . . . . 11 | |
13 | 6, 11, 12 | syl2an 463 | . . . . . . . . . 10 |
14 | 13 | exp43 595 | . . . . . . . . 9 |
15 | 14 | com23 72 | . . . . . . . 8 |
16 | 15 | imp 418 | . . . . . . 7 |
17 | 16 | com34 77 | . . . . . 6 |
18 | 17 | pm2.43d 44 | . . . . 5 |
19 | 18 | imp3a 420 | . . . 4 |
20 | 5, 19 | impbid 183 | . . 3 |
21 | opelres 4951 | . . 3 | |
22 | 20, 21 | syl6rbbr 255 | . 2 |
23 | 22 | eqrelrdv 4853 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 wex 1541 wceq 1642 wcel 1710 weu 2204 wss 3258 cop 4562 cdm 4773 cres 4775 wfun 4776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-reu 2622 df-rmo 2623 df-rab 2624 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-pss 3262 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-0c 4378 df-addc 4379 df-nnc 4380 df-fin 4381 df-lefin 4441 df-ltfin 4442 df-ncfin 4443 df-tfin 4444 df-evenfin 4445 df-oddfin 4446 df-sfin 4447 df-spfin 4448 df-phi 4566 df-op 4567 df-proj1 4568 df-proj2 4569 df-opab 4624 df-br 4641 df-co 4727 df-ima 4728 df-id 4768 df-xp 4785 df-cnv 4786 df-rn 4787 df-dm 4788 df-res 4789 df-fun 4790 |
This theorem is referenced by: fun2ssres 5146 funcnvres 5166 funssfv 5344 oprssov 5604 |
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