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Mirrors > Home > ILE Home > Th. List > ressbasd | GIF version |
Description: Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.) |
Ref | Expression |
---|---|
ressbasd.r | β’ (π β π = (π βΎs π΄)) |
ressbasd.b | β’ (π β π΅ = (Baseβπ)) |
ressbasd.w | β’ (π β π β π) |
ressbasd.a | β’ (π β π΄ β π) |
Ref | Expression |
---|---|
ressbasd | β’ (π β (π΄ β© π΅) = (Baseβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressbasd.w | . . 3 β’ (π β π β π) | |
2 | ressbasd.a | . . . 4 β’ (π β π΄ β π) | |
3 | inex1g 4139 | . . . 4 β’ (π΄ β π β (π΄ β© (Baseβπ)) β V) | |
4 | 2, 3 | syl 14 | . . 3 β’ (π β (π΄ β© (Baseβπ)) β V) |
5 | baseslid 12518 | . . . 4 β’ (Base = Slot (Baseβndx) β§ (Baseβndx) β β) | |
6 | 5 | setsslid 12512 | . . 3 β’ ((π β π β§ (π΄ β© (Baseβπ)) β V) β (π΄ β© (Baseβπ)) = (Baseβ(π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©))) |
7 | 1, 4, 6 | syl2anc 411 | . 2 β’ (π β (π΄ β© (Baseβπ)) = (Baseβ(π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©))) |
8 | ressbasd.b | . . 3 β’ (π β π΅ = (Baseβπ)) | |
9 | 8 | ineq2d 3336 | . 2 β’ (π β (π΄ β© π΅) = (π΄ β© (Baseβπ))) |
10 | ressbasd.r | . . . 4 β’ (π β π = (π βΎs π΄)) | |
11 | ressvalsets 12523 | . . . . 5 β’ ((π β π β§ π΄ β π) β (π βΎs π΄) = (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) | |
12 | 1, 2, 11 | syl2anc 411 | . . . 4 β’ (π β (π βΎs π΄) = (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) |
13 | 10, 12 | eqtrd 2210 | . . 3 β’ (π β π = (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) |
14 | 13 | fveq2d 5519 | . 2 β’ (π β (Baseβπ ) = (Baseβ(π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©))) |
15 | 7, 9, 14 | 3eqtr4d 2220 | 1 β’ (π β (π΄ β© π΅) = (Baseβπ )) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 Vcvv 2737 β© cin 3128 β¨cop 3595 βcfv 5216 (class class class)co 5874 ndxcnx 12458 sSet csts 12459 Basecbs 12461 βΎs cress 12462 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-inn 8919 df-ndx 12464 df-slot 12465 df-base 12467 df-sets 12468 df-iress 12469 |
This theorem is referenced by: ressbas2d 12527 ressbasssd 12528 ressressg 12533 grpressid 12930 subrgpropd 13367 |
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