![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ressval3d | GIF version |
Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.) |
Ref | Expression |
---|---|
ressval3d.r | β’ π = (π βΎs π΄) |
ressval3d.b | β’ π΅ = (Baseβπ) |
ressval3d.e | β’ πΈ = (Baseβndx) |
ressval3d.s | β’ (π β π β π) |
ressval3d.f | β’ (π β Fun π) |
ressval3d.d | β’ (π β πΈ β dom π) |
ressval3d.u | β’ (π β π΄ β π΅) |
Ref | Expression |
---|---|
ressval3d | β’ (π β π = (π sSet β¨πΈ, π΄β©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressval3d.r | . . 3 β’ π = (π βΎs π΄) | |
2 | ressval3d.s | . . . 4 β’ (π β π β π) | |
3 | ressval3d.b | . . . . . 6 β’ π΅ = (Baseβπ) | |
4 | basfn 12533 | . . . . . . 7 β’ Base Fn V | |
5 | 2 | elexd 2762 | . . . . . . 7 β’ (π β π β V) |
6 | funfvex 5544 | . . . . . . . 8 β’ ((Fun Base β§ π β dom Base) β (Baseβπ) β V) | |
7 | 6 | funfni 5328 | . . . . . . 7 β’ ((Base Fn V β§ π β V) β (Baseβπ) β V) |
8 | 4, 5, 7 | sylancr 414 | . . . . . 6 β’ (π β (Baseβπ) β V) |
9 | 3, 8 | eqeltrid 2274 | . . . . 5 β’ (π β π΅ β V) |
10 | ressval3d.u | . . . . 5 β’ (π β π΄ β π΅) | |
11 | 9, 10 | ssexd 4155 | . . . 4 β’ (π β π΄ β V) |
12 | ressvalsets 12537 | . . . 4 β’ ((π β π β§ π΄ β V) β (π βΎs π΄) = (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) | |
13 | 2, 11, 12 | syl2anc 411 | . . 3 β’ (π β (π βΎs π΄) = (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) |
14 | 1, 13 | eqtrid 2232 | . 2 β’ (π β π = (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) |
15 | ressval3d.e | . . . . 5 β’ πΈ = (Baseβndx) | |
16 | 15 | a1i 9 | . . . 4 β’ (π β πΈ = (Baseβndx)) |
17 | df-ss 3154 | . . . . . 6 β’ (π΄ β π΅ β (π΄ β© π΅) = π΄) | |
18 | 10, 17 | sylib 122 | . . . . 5 β’ (π β (π΄ β© π΅) = π΄) |
19 | 3 | ineq2i 3345 | . . . . 5 β’ (π΄ β© π΅) = (π΄ β© (Baseβπ)) |
20 | 18, 19 | eqtr3di 2235 | . . . 4 β’ (π β π΄ = (π΄ β© (Baseβπ))) |
21 | 16, 20 | opeq12d 3798 | . . 3 β’ (π β β¨πΈ, π΄β© = β¨(Baseβndx), (π΄ β© (Baseβπ))β©) |
22 | 21 | oveq2d 5904 | . 2 β’ (π β (π sSet β¨πΈ, π΄β©) = (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) |
23 | 14, 22 | eqtr4d 2223 | 1 β’ (π β π = (π sSet β¨πΈ, π΄β©)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1363 β wcel 2158 Vcvv 2749 β© cin 3140 β wss 3141 β¨cop 3607 dom cdm 4638 Fun wfun 5222 Fn wfn 5223 βcfv 5228 (class class class)co 5888 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-1re 7918 ax-addrcl 7921 |
This theorem depends on definitions: df-bi 117 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-ral 2470 df-rex 2471 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-iota 5190 df-fun 5230 df-fn 5231 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-inn 8933 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 |