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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 17185 | . . 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 4532 | . . 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 2788 | . 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 1112 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β© cin 3942 β wss 3943 ifcif 4523 β¨cop 4629 βcfv 6537 (class class class)co 7405 sSet csts 17105 ndxcnx 17135 Basecbs 17153 βΎs cress 17182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-ress 17183 |
This theorem is referenced by: ressbas 17188 ressbasOLD 17189 resseqnbas 17195 resslemOLD 17196 ressinbas 17199 ressval3d 17200 ressval3dOLD 17201 ressress 17202 rescabs 17791 rescabsOLD 17792 symgvalstruct 19316 symgvalstructOLD 19317 mgpress 20054 mgpressOLD 20055 resssra 33192 |
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