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Mirrors > Home > MPE Home > Th. List > ressval2 | Structured version Visualization version GIF version |
Description: Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
Ref | Expression |
---|---|
ressbas.r | β’ π = (π βΎs π΄) |
ressbas.b | β’ π΅ = (Baseβπ) |
Ref | Expression |
---|---|
ressval2 | β’ ((Β¬ π΅ β π΄ β§ π β π β§ π΄ β π) β π = (π sSet β¨(Baseβndx), (π΄ β© π΅)β©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressbas.r | . . . 4 β’ π = (π βΎs π΄) | |
2 | ressbas.b | . . . 4 β’ π΅ = (Baseβπ) | |
3 | 1, 2 | ressval 17120 | . . 3 β’ ((π β π β§ π΄ β π) β π = if(π΅ β π΄, π, (π sSet β¨(Baseβndx), (π΄ β© π΅)β©))) |
4 | iffalse 4496 | . . 3 β’ (Β¬ π΅ β π΄ β if(π΅ β π΄, π, (π sSet β¨(Baseβndx), (π΄ β© π΅)β©)) = (π sSet β¨(Baseβndx), (π΄ β© π΅)β©)) | |
5 | 3, 4 | sylan9eqr 2795 | . 2 β’ ((Β¬ π΅ β π΄ β§ (π β π β§ π΄ β π)) β π = (π sSet β¨(Baseβndx), (π΄ β© π΅)β©)) |
6 | 5 | 3impb 1116 | 1 β’ ((Β¬ π΅ β π΄ β§ π β π β§ π΄ β π) β π = (π sSet β¨(Baseβndx), (π΄ β© π΅)β©)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β© cin 3910 β wss 3911 ifcif 4487 β¨cop 4593 βcfv 6497 (class class class)co 7358 sSet csts 17040 ndxcnx 17070 Basecbs 17088 βΎs cress 17117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-ress 17118 |
This theorem is referenced by: ressbas 17123 ressbasOLD 17124 resseqnbas 17127 resslemOLD 17128 ressinbas 17131 ressval3d 17132 ressval3dOLD 17133 ressress 17134 rescabs 17723 rescabsOLD 17724 symgvalstruct 19183 symgvalstructOLD 19184 mgpress 19916 mgpressOLD 19917 |
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