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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 983 | . . . . 5 | |
3 | 2 | elexd 2725 | . . . 4 |
4 | simp3 984 | . . . . 5 | |
5 | 4 | elexd 2725 | . . . 4 |
6 | simp1 982 | . . . . . 6 | |
7 | 6 | iftrued 3513 | . . . . 5 sSet |
8 | 7, 3 | eqeltrd 2234 | . . . 4 sSet |
9 | simpl 108 | . . . . . . . . 9 | |
10 | 9 | fveq2d 5474 | . . . . . . . 8 |
11 | ressbas.b | . . . . . . . 8 | |
12 | 10, 11 | eqtr4di 2208 | . . . . . . 7 |
13 | simpr 109 | . . . . . . 7 | |
14 | 12, 13 | sseq12d 3159 | . . . . . 6 |
15 | 13, 12 | ineq12d 3310 | . . . . . . . 8 |
16 | 15 | opeq2d 3750 | . . . . . . 7 |
17 | 9, 16 | oveq12d 5844 | . . . . . 6 sSet sSet |
18 | 14, 9, 17 | ifbieq12d 3532 | . . . . 5 sSet sSet |
19 | df-ress 12268 | . . . . 5 ↾s sSet | |
20 | 18, 19 | ovmpoga 5952 | . . . 4 sSet ↾s sSet |
21 | 3, 5, 8, 20 | syl3anc 1220 | . . 3 ↾s sSet |
22 | 21, 7 | eqtrd 2190 | . 2 ↾s |
23 | 1, 22 | syl5eq 2202 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wceq 1335 wcel 2128 cvv 2712 cin 3101 wss 3102 cif 3506 cop 3564 cfv 5172 (class class class)co 5826 cnx 12257 sSet csts 12258 cbs 12260 ↾s cress 12261 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-setind 4498 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-ress 12268 |
This theorem is referenced by: ressid 12321 |
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