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Mirrors > Home > ILE Home > Th. List > ressid2 | Unicode version |
Description: General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 26-Jan-2023.) |
Ref | Expression |
---|---|
ressbas.r | ↾s |
ressbas.b |
Ref | Expression |
---|---|
ressid2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressbas.r | . 2 ↾s | |
2 | simp2 967 | . . . . 5 | |
3 | 2 | elexd 2673 | . . . 4 |
4 | simp3 968 | . . . . 5 | |
5 | 4 | elexd 2673 | . . . 4 |
6 | simp1 966 | . . . . . 6 | |
7 | 6 | iftrued 3451 | . . . . 5 sSet |
8 | 7, 3 | eqeltrd 2194 | . . . 4 sSet |
9 | simpl 108 | . . . . . . . . 9 | |
10 | 9 | fveq2d 5393 | . . . . . . . 8 |
11 | ressbas.b | . . . . . . . 8 | |
12 | 10, 11 | syl6eqr 2168 | . . . . . . 7 |
13 | simpr 109 | . . . . . . 7 | |
14 | 12, 13 | sseq12d 3098 | . . . . . 6 |
15 | 13, 12 | ineq12d 3248 | . . . . . . . 8 |
16 | 15 | opeq2d 3682 | . . . . . . 7 |
17 | 9, 16 | oveq12d 5760 | . . . . . 6 sSet sSet |
18 | 14, 9, 17 | ifbieq12d 3468 | . . . . 5 sSet sSet |
19 | df-ress 11894 | . . . . 5 ↾s sSet | |
20 | 18, 19 | ovmpoga 5868 | . . . 4 sSet ↾s sSet |
21 | 3, 5, 8, 20 | syl3anc 1201 | . . 3 ↾s sSet |
22 | 21, 7 | eqtrd 2150 | . 2 ↾s |
23 | 1, 22 | syl5eq 2162 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 cvv 2660 cin 3040 wss 3041 cif 3444 cop 3500 cfv 5093 (class class class)co 5742 cnx 11883 sSet csts 11884 cbs 11886 ↾s cress 11887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-ress 11894 |
This theorem is referenced by: ressid 11947 |
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