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Mirrors > Home > ILE Home > Th. List > setsslid | Unicode version |
Description: Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.) |
Ref | Expression |
---|---|
setsslid.e | Slot |
Ref | Expression |
---|---|
setsslid | sSet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsslid.e | . . . . 5 Slot | |
2 | 1 | simpri 112 | . . . 4 |
3 | setsvala 12291 | . . . 4 sSet | |
4 | 2, 3 | mp3an2 1307 | . . 3 sSet |
5 | 4 | fveq2d 5474 | . 2 sSet |
6 | 1 | simpli 110 | . . 3 Slot |
7 | resexg 4908 | . . . 4 | |
8 | simpr 109 | . . . . . 6 | |
9 | opexg 4190 | . . . . . 6 | |
10 | 2, 8, 9 | sylancr 411 | . . . . 5 |
11 | snexg 4147 | . . . . 5 | |
12 | 10, 11 | syl 14 | . . . 4 |
13 | unexg 4405 | . . . 4 | |
14 | 7, 12, 13 | syl2an2r 585 | . . 3 |
15 | 2 | a1i 9 | . . 3 |
16 | 6, 14, 15 | strnfvnd 12280 | . 2 |
17 | snidg 3590 | . . . . 5 | |
18 | fvres 5494 | . . . . 5 | |
19 | 2, 17, 18 | mp2b 8 | . . . 4 |
20 | resres 4880 | . . . . . . . . 9 | |
21 | incom 3300 | . . . . . . . . . . . 12 | |
22 | disjdif 3467 | . . . . . . . . . . . 12 | |
23 | 21, 22 | eqtri 2178 | . . . . . . . . . . 11 |
24 | 23 | reseq2i 4865 | . . . . . . . . . 10 |
25 | res0 4872 | . . . . . . . . . 10 | |
26 | 24, 25 | eqtri 2178 | . . . . . . . . 9 |
27 | 20, 26 | eqtri 2178 | . . . . . . . 8 |
28 | 27 | a1i 9 | . . . . . . 7 |
29 | 2 | elexi 2724 | . . . . . . . . . 10 |
30 | 8 | elexd 2725 | . . . . . . . . . 10 |
31 | opelxpi 4620 | . . . . . . . . . 10 | |
32 | 29, 30, 31 | sylancr 411 | . . . . . . . . 9 |
33 | relsng 4691 | . . . . . . . . . 10 | |
34 | 10, 33 | syl 14 | . . . . . . . . 9 |
35 | 32, 34 | mpbird 166 | . . . . . . . 8 |
36 | dmsnopg 5059 | . . . . . . . . . 10 | |
37 | 36 | adantl 275 | . . . . . . . . 9 |
38 | eqimss 3182 | . . . . . . . . 9 | |
39 | 37, 38 | syl 14 | . . . . . . . 8 |
40 | relssres 4906 | . . . . . . . 8 | |
41 | 35, 39, 40 | syl2anc 409 | . . . . . . 7 |
42 | 28, 41 | uneq12d 3263 | . . . . . 6 |
43 | resundir 4882 | . . . . . 6 | |
44 | un0 3428 | . . . . . . 7 | |
45 | uncom 3252 | . . . . . . 7 | |
46 | 44, 45 | eqtr3i 2180 | . . . . . 6 |
47 | 42, 43, 46 | 3eqtr4g 2215 | . . . . 5 |
48 | 47 | fveq1d 5472 | . . . 4 |
49 | 19, 48 | eqtr3id 2204 | . . 3 |
50 | fvsng 5665 | . . . 4 | |
51 | 2, 8, 50 | sylancr 411 | . . 3 |
52 | 49, 51 | eqtrd 2190 | . 2 |
53 | 5, 16, 52 | 3eqtrrd 2195 | 1 sSet |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 cvv 2712 cdif 3099 cun 3100 cin 3101 wss 3102 c0 3395 csn 3561 cop 3564 cxp 4586 cdm 4588 cres 4590 wrel 4593 cfv 5172 (class class class)co 5826 cn 8838 cnx 12257 sSet csts 12258 Slot cslot 12259 |
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-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 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-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-iota 5137 df-fun 5174 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-slot 12264 df-sets 12267 |
This theorem is referenced by: setsmstsetg 12951 |
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