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Mirrors > Home > ILE Home > Th. List > ressinbasd | GIF version |
Description: Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
Ref | Expression |
---|---|
ressidbasd.1 | β’ (π β π΅ = (Baseβπ)) |
ressidbasd.a | β’ (π β π΄ β π) |
ressidbasd.w | β’ (π β π β π) |
Ref | Expression |
---|---|
ressinbasd | β’ (π β (π βΎs π΄) = (π βΎs (π΄ β© π΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressidbasd.1 | . . . . . . 7 β’ (π β π΅ = (Baseβπ)) | |
2 | inidm 3359 | . . . . . . . 8 β’ (π΅ β© π΅) = π΅ | |
3 | 1 | ineq2d 3351 | . . . . . . . 8 β’ (π β (π΅ β© π΅) = (π΅ β© (Baseβπ))) |
4 | 2, 3 | eqtr3id 2236 | . . . . . . 7 β’ (π β π΅ = (π΅ β© (Baseβπ))) |
5 | 1, 4 | eqtr3d 2224 | . . . . . 6 β’ (π β (Baseβπ) = (π΅ β© (Baseβπ))) |
6 | 5 | ineq2d 3351 | . . . . 5 β’ (π β (π΄ β© (Baseβπ)) = (π΄ β© (π΅ β© (Baseβπ)))) |
7 | inass 3360 | . . . . 5 β’ ((π΄ β© π΅) β© (Baseβπ)) = (π΄ β© (π΅ β© (Baseβπ))) | |
8 | 6, 7 | eqtr4di 2240 | . . . 4 β’ (π β (π΄ β© (Baseβπ)) = ((π΄ β© π΅) β© (Baseβπ))) |
9 | 8 | opeq2d 3800 | . . 3 β’ (π β β¨(Baseβndx), (π΄ β© (Baseβπ))β© = β¨(Baseβndx), ((π΄ β© π΅) β© (Baseβπ))β©) |
10 | 9 | oveq2d 5907 | . 2 β’ (π β (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©) = (π sSet β¨(Baseβndx), ((π΄ β© π΅) β© (Baseβπ))β©)) |
11 | ressidbasd.w | . . 3 β’ (π β π β π) | |
12 | ressidbasd.a | . . 3 β’ (π β π΄ β π) | |
13 | ressvalsets 12542 | . . 3 β’ ((π β π β§ π΄ β π) β (π βΎs π΄) = (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) | |
14 | 11, 12, 13 | syl2anc 411 | . 2 β’ (π β (π βΎs π΄) = (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) |
15 | inex1g 4154 | . . . 4 β’ (π΄ β π β (π΄ β© π΅) β V) | |
16 | 12, 15 | syl 14 | . . 3 β’ (π β (π΄ β© π΅) β V) |
17 | ressvalsets 12542 | . . 3 β’ ((π β π β§ (π΄ β© π΅) β V) β (π βΎs (π΄ β© π΅)) = (π sSet β¨(Baseβndx), ((π΄ β© π΅) β© (Baseβπ))β©)) | |
18 | 11, 16, 17 | syl2anc 411 | . 2 β’ (π β (π βΎs (π΄ β© π΅)) = (π sSet β¨(Baseβndx), ((π΄ β© π΅) β© (Baseβπ))β©)) |
19 | 10, 14, 18 | 3eqtr4d 2232 | 1 β’ (π β (π βΎs π΄) = (π βΎs (π΄ β© π΅))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1364 β wcel 2160 Vcvv 2752 β© cin 3143 β¨cop 3610 βcfv 5231 (class class class)co 5891 ndxcnx 12477 sSet csts 12478 Basecbs 12480 βΎs cress 12481 |
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 7920 ax-resscn 7921 ax-1re 7923 ax-addrcl 7926 |
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-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-inn 8938 df-ndx 12483 df-slot 12484 df-base 12486 df-sets 12487 df-iress 12488 |
This theorem is referenced by: (None) |
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