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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 3356 | . . . . . . . 8 β’ (π΅ β© π΅) = π΅ | |
3 | 1 | ineq2d 3348 | . . . . . . . 8 β’ (π β (π΅ β© π΅) = (π΅ β© (Baseβπ))) |
4 | 2, 3 | eqtr3id 2234 | . . . . . . 7 β’ (π β π΅ = (π΅ β© (Baseβπ))) |
5 | 1, 4 | eqtr3d 2222 | . . . . . 6 β’ (π β (Baseβπ) = (π΅ β© (Baseβπ))) |
6 | 5 | ineq2d 3348 | . . . . 5 β’ (π β (π΄ β© (Baseβπ)) = (π΄ β© (π΅ β© (Baseβπ)))) |
7 | inass 3357 | . . . . 5 β’ ((π΄ β© π΅) β© (Baseβπ)) = (π΄ β© (π΅ β© (Baseβπ))) | |
8 | 6, 7 | eqtr4di 2238 | . . . 4 β’ (π β (π΄ β© (Baseβπ)) = ((π΄ β© π΅) β© (Baseβπ))) |
9 | 8 | opeq2d 3797 | . . 3 β’ (π β β¨(Baseβndx), (π΄ β© (Baseβπ))β© = β¨(Baseβndx), ((π΄ β© π΅) β© (Baseβπ))β©) |
10 | 9 | oveq2d 5904 | . 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 12537 | . . 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 4151 | . . . 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 12537 | . . 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 2230 | 1 β’ (π β (π βΎs π΄) = (π βΎs (π΄ β© 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 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-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) |
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