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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 993 | . . . . 5 | |
3 | 2 | elexd 2743 | . . . 4 |
4 | simp3 994 | . . . . 5 | |
5 | 4 | elexd 2743 | . . . 4 |
6 | simp1 992 | . . . . . 6 | |
7 | 6 | iftrued 3533 | . . . . 5 sSet |
8 | 7, 3 | eqeltrd 2247 | . . . 4 sSet |
9 | simpl 108 | . . . . . . . . 9 | |
10 | 9 | fveq2d 5500 | . . . . . . . 8 |
11 | ressbas.b | . . . . . . . 8 | |
12 | 10, 11 | eqtr4di 2221 | . . . . . . 7 |
13 | simpr 109 | . . . . . . 7 | |
14 | 12, 13 | sseq12d 3178 | . . . . . 6 |
15 | 13, 12 | ineq12d 3329 | . . . . . . . 8 |
16 | 15 | opeq2d 3772 | . . . . . . 7 |
17 | 9, 16 | oveq12d 5871 | . . . . . 6 sSet sSet |
18 | 14, 9, 17 | ifbieq12d 3552 | . . . . 5 sSet sSet |
19 | df-ress 12424 | . . . . 5 ↾s sSet | |
20 | 18, 19 | ovmpoga 5982 | . . . 4 sSet ↾s sSet |
21 | 3, 5, 8, 20 | syl3anc 1233 | . . 3 ↾s sSet |
22 | 21, 7 | eqtrd 2203 | . 2 ↾s |
23 | 1, 22 | eqtrid 2215 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 cvv 2730 cin 3120 wss 3121 cif 3526 cop 3586 cfv 5198 (class class class)co 5853 cnx 12413 sSet csts 12414 cbs 12416 ↾s cress 12417 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-ress 12424 |
This theorem is referenced by: ressid 12479 |
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