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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 12544 | . . . . . . 7 β’ Base Fn V | |
5 | 2 | elexd 2765 | . . . . . . 7 β’ (π β π β V) |
6 | funfvex 5547 | . . . . . . . 8 β’ ((Fun Base β§ π β dom Base) β (Baseβπ) β V) | |
7 | 6 | funfni 5331 | . . . . . . 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 2276 | . . . . 5 β’ (π β π΅ β V) |
10 | ressval3d.u | . . . . 5 β’ (π β π΄ β π΅) | |
11 | 9, 10 | ssexd 4158 | . . . 4 β’ (π β π΄ β V) |
12 | ressvalsets 12548 | . . . 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 2234 | . 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 3157 | . . . . . 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 3348 | . . . . 5 β’ (π΄ β© π΅) = (π΄ β© (Baseβπ)) |
20 | 18, 19 | eqtr3di 2237 | . . . 4 β’ (π β π΄ = (π΄ β© (Baseβπ))) |
21 | 16, 20 | opeq12d 3801 | . . 3 β’ (π β β¨πΈ, π΄β© = β¨(Baseβndx), (π΄ β© (Baseβπ))β©) |
22 | 21 | oveq2d 5907 | . 2 β’ (π β (π sSet β¨πΈ, π΄β©) = (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) |
23 | 14, 22 | eqtr4d 2225 | 1 β’ (π β π = (π sSet β¨πΈ, π΄β©)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1364 β wcel 2160 Vcvv 2752 β© cin 3143 β wss 3144 β¨cop 3610 dom cdm 4641 Fun wfun 5225 Fn wfn 5226 βcfv 5231 (class class class)co 5891 ndxcnx 12483 sSet csts 12484 Basecbs 12486 βΎs cress 12487 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1re 7924 ax-addrcl 7927 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-iota 5193 df-fun 5233 df-fn 5234 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-inn 8939 df-ndx 12489 df-slot 12490 df-base 12492 df-sets 12493 df-iress 12494 |
This theorem is referenced by: (None) |
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