![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > strressid | GIF version |
Description: Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.) |
Ref | Expression |
---|---|
strressid.b | β’ (π β π΅ = (Baseβπ)) |
strressid.s | β’ (π β π Struct β¨π, πβ©) |
strressid.f | β’ (π β Fun π) |
strressid.bw | β’ (π β (Baseβndx) β dom π) |
Ref | Expression |
---|---|
strressid | β’ (π β (π βΎs π΅) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strressid.b | . . . . . 6 β’ (π β π΅ = (Baseβπ)) | |
2 | 1 | ineq1d 3347 | . . . . 5 β’ (π β (π΅ β© (Baseβπ)) = ((Baseβπ) β© (Baseβπ))) |
3 | inidm 3356 | . . . . 5 β’ ((Baseβπ) β© (Baseβπ)) = (Baseβπ) | |
4 | 2, 3 | eqtrdi 2236 | . . . 4 β’ (π β (π΅ β© (Baseβπ)) = (Baseβπ)) |
5 | 4 | opeq2d 3797 | . . 3 β’ (π β β¨(Baseβndx), (π΅ β© (Baseβπ))β© = β¨(Baseβndx), (Baseβπ)β©) |
6 | 5 | oveq2d 5904 | . 2 β’ (π β (π sSet β¨(Baseβndx), (π΅ β© (Baseβπ))β©) = (π sSet β¨(Baseβndx), (Baseβπ)β©)) |
7 | strressid.s | . . . 4 β’ (π β π Struct β¨π, πβ©) | |
8 | structex 12487 | . . . 4 β’ (π Struct β¨π, πβ© β π β V) | |
9 | 7, 8 | syl 14 | . . 3 β’ (π β π β V) |
10 | basfn 12533 | . . . . 5 β’ Base Fn V | |
11 | funfvex 5544 | . . . . . 6 β’ ((Fun Base β§ π β dom Base) β (Baseβπ) β V) | |
12 | 11 | funfni 5328 | . . . . 5 β’ ((Base Fn V β§ π β V) β (Baseβπ) β V) |
13 | 10, 9, 12 | sylancr 414 | . . . 4 β’ (π β (Baseβπ) β V) |
14 | 1, 13 | eqeltrd 2264 | . . 3 β’ (π β π΅ β V) |
15 | ressvalsets 12537 | . . 3 β’ ((π β V β§ π΅ β V) β (π βΎs π΅) = (π sSet β¨(Baseβndx), (π΅ β© (Baseβπ))β©)) | |
16 | 9, 14, 15 | syl2anc 411 | . 2 β’ (π β (π βΎs π΅) = (π sSet β¨(Baseβndx), (π΅ β© (Baseβπ))β©)) |
17 | baseid 12529 | . . 3 β’ Base = Slot (Baseβndx) | |
18 | strressid.f | . . 3 β’ (π β Fun π) | |
19 | strressid.bw | . . 3 β’ (π β (Baseβndx) β dom π) | |
20 | 17, 7, 18, 19 | strsetsid 12508 | . 2 β’ (π β π = (π sSet β¨(Baseβndx), (Baseβπ)β©)) |
21 | 6, 16, 20 | 3eqtr4d 2230 | 1 β’ (π β (π βΎs π΅) = π) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1363 β wcel 2158 Vcvv 2749 β© cin 3140 β¨cop 3607 class class class wbr 4015 dom cdm 4638 Fun wfun 5222 Fn wfn 5223 βcfv 5228 (class class class)co 5888 Struct cstr 12471 ndxcnx 12472 sSet csts 12473 Basecbs 12475 βΎs cress 12476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-inn 8933 df-n0 9190 df-z 9267 df-uz 9542 df-fz 10022 df-struct 12477 df-ndx 12478 df-slot 12479 df-base 12481 df-sets 12482 df-iress 12483 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |