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Mirrors > Home > ILE Home > Th. List > iss | Unicode version |
Description: A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
iss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2939 | . . . . . . 7 | |
2 | vex 2560 | . . . . . . . . 9 | |
3 | vex 2560 | . . . . . . . . 9 | |
4 | 2, 3 | opeldm 4538 | . . . . . . . 8 |
5 | 4 | a1i 9 | . . . . . . 7 |
6 | 1, 5 | jcad 291 | . . . . . 6 |
7 | df-br 3765 | . . . . . . . . 9 | |
8 | 3 | ideq 4488 | . . . . . . . . 9 |
9 | 7, 8 | bitr3i 175 | . . . . . . . 8 |
10 | 2 | eldm2 4533 | . . . . . . . . . 10 |
11 | opeq2 3550 | . . . . . . . . . . . . . . 15 | |
12 | 11 | eleq1d 2106 | . . . . . . . . . . . . . 14 |
13 | 12 | biimprcd 149 | . . . . . . . . . . . . 13 |
14 | 9, 13 | syl5bi 141 | . . . . . . . . . . . 12 |
15 | 1, 14 | sylcom 25 | . . . . . . . . . . 11 |
16 | 15 | exlimdv 1700 | . . . . . . . . . 10 |
17 | 10, 16 | syl5bi 141 | . . . . . . . . 9 |
18 | 12 | imbi2d 219 | . . . . . . . . 9 |
19 | 17, 18 | syl5ibcom 144 | . . . . . . . 8 |
20 | 9, 19 | syl5bi 141 | . . . . . . 7 |
21 | 20 | impd 242 | . . . . . 6 |
22 | 6, 21 | impbid 120 | . . . . 5 |
23 | 3 | opelres 4617 | . . . . 5 |
24 | 22, 23 | syl6bbr 187 | . . . 4 |
25 | 24 | alrimivv 1755 | . . 3 |
26 | reli 4465 | . . . . 5 | |
27 | relss 4427 | . . . . 5 | |
28 | 26, 27 | mpi 15 | . . . 4 |
29 | relres 4639 | . . . 4 | |
30 | eqrel 4429 | . . . 4 | |
31 | 28, 29, 30 | sylancl 392 | . . 3 |
32 | 25, 31 | mpbird 156 | . 2 |
33 | resss 4635 | . . 3 | |
34 | sseq1 2966 | . . 3 | |
35 | 33, 34 | mpbiri 157 | . 2 |
36 | 32, 35 | impbii 117 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wal 1241 wceq 1243 wex 1381 wcel 1393 wss 2917 cop 3378 class class class wbr 3764 cid 4025 cdm 4345 cres 4347 wrel 4350 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-dm 4355 df-res 4357 |
This theorem is referenced by: funcocnv2 5151 |
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