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| Mirrors > Home > MPE Home > Th. List > ressid2 | Structured version Visualization version GIF version | ||
| Description: General behavior of trivial 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 |
|---|---|
| ressid2 | β’ ((π΅ β π΄ β§ π β π β§ π΄ β π) β π = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ressbas.r | . . . 4 β’ π = (π βΎs π΄) | |
| 2 | ressbas.b | . . . 4 β’ π΅ = (Baseβπ) | |
| 3 | 1, 2 | ressval 17176 | . . 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 | iftrue 4535 | . . 3 β’ (π΅ β π΄ β if(π΅ β π΄, π, (π sSet β¨(Baseβndx), (π΄ β© π΅)β©)) = π) | |
| 5 | 3, 4 | sylan9eqr 2795 | . 2 β’ ((π΅ β π΄ β§ (π β π β§ π΄ β π)) β π = π) |
| 6 | 5 | 3impb 1116 | 1 β’ ((π΅ β π΄ β§ π β π β§ π΄ β π) β π = π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β© cin 3948 β wss 3949 ifcif 4529 β¨cop 4635 βcfv 6544 (class class class)co 7409 sSet csts 17096 ndxcnx 17126 Basecbs 17144 βΎs cress 17173 |
| 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 5300 ax-nul 5307 ax-pr 5428 |
| 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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-ress 17174 |
| This theorem is referenced by: ressbas 17179 ressbasOLD 17180 resseqnbas 17186 resslemOLD 17187 ress0 17188 ressid 17189 ressinbas 17190 ressress 17193 rescabs 17782 rescabsOLD 17783 0symgefmndeq 19261 snsymgefmndeq 19262 mgpress 20002 mgpressOLD 20003 psgnghm2 21134 resvsca 32444 extdg1id 32742 |
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