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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 12447 | . . . 4 sSet | |
4 | 2, 3 | mp3an2 1320 | . . 3 sSet |
5 | 4 | fveq2d 5500 | . 2 sSet |
6 | 1 | simpli 110 | . . 3 Slot |
7 | resexg 4931 | . . . 4 | |
8 | simpr 109 | . . . . . 6 | |
9 | opexg 4213 | . . . . . 6 | |
10 | 2, 8, 9 | sylancr 412 | . . . . 5 |
11 | snexg 4170 | . . . . 5 | |
12 | 10, 11 | syl 14 | . . . 4 |
13 | unexg 4428 | . . . 4 | |
14 | 7, 12, 13 | syl2an2r 590 | . . 3 |
15 | 2 | a1i 9 | . . 3 |
16 | 6, 14, 15 | strnfvnd 12436 | . 2 |
17 | snidg 3612 | . . . . 5 | |
18 | fvres 5520 | . . . . 5 | |
19 | 2, 17, 18 | mp2b 8 | . . . 4 |
20 | resres 4903 | . . . . . . . . 9 | |
21 | incom 3319 | . . . . . . . . . . . 12 | |
22 | disjdif 3487 | . . . . . . . . . . . 12 | |
23 | 21, 22 | eqtri 2191 | . . . . . . . . . . 11 |
24 | 23 | reseq2i 4888 | . . . . . . . . . 10 |
25 | res0 4895 | . . . . . . . . . 10 | |
26 | 24, 25 | eqtri 2191 | . . . . . . . . 9 |
27 | 20, 26 | eqtri 2191 | . . . . . . . 8 |
28 | 27 | a1i 9 | . . . . . . 7 |
29 | 2 | elexi 2742 | . . . . . . . . . 10 |
30 | 8 | elexd 2743 | . . . . . . . . . 10 |
31 | opelxpi 4643 | . . . . . . . . . 10 | |
32 | 29, 30, 31 | sylancr 412 | . . . . . . . . 9 |
33 | relsng 4714 | . . . . . . . . . 10 | |
34 | 10, 33 | syl 14 | . . . . . . . . 9 |
35 | 32, 34 | mpbird 166 | . . . . . . . 8 |
36 | dmsnopg 5082 | . . . . . . . . . 10 | |
37 | 36 | adantl 275 | . . . . . . . . 9 |
38 | eqimss 3201 | . . . . . . . . 9 | |
39 | 37, 38 | syl 14 | . . . . . . . 8 |
40 | relssres 4929 | . . . . . . . 8 | |
41 | 35, 39, 40 | syl2anc 409 | . . . . . . 7 |
42 | 28, 41 | uneq12d 3282 | . . . . . 6 |
43 | resundir 4905 | . . . . . 6 | |
44 | un0 3448 | . . . . . . 7 | |
45 | uncom 3271 | . . . . . . 7 | |
46 | 44, 45 | eqtr3i 2193 | . . . . . 6 |
47 | 42, 43, 46 | 3eqtr4g 2228 | . . . . 5 |
48 | 47 | fveq1d 5498 | . . . 4 |
49 | 19, 48 | eqtr3id 2217 | . . 3 |
50 | fvsng 5692 | . . . 4 | |
51 | 2, 8, 50 | sylancr 412 | . . 3 |
52 | 49, 51 | eqtrd 2203 | . 2 |
53 | 5, 16, 52 | 3eqtrrd 2208 | 1 sSet |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 cvv 2730 cdif 3118 cun 3119 cin 3120 wss 3121 c0 3414 csn 3583 cop 3586 cxp 4609 cdm 4611 cres 4613 wrel 4616 cfv 5198 (class class class)co 5853 cn 8878 cnx 12413 sSet csts 12414 Slot cslot 12415 |
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-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 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-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-slot 12420 df-sets 12423 |
This theorem is referenced by: setsmstsetg 13275 |
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