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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 17182 | . . 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 4529 | . . 3 β’ (π΅ β π΄ β if(π΅ β π΄, π, (π sSet β¨(Baseβndx), (π΄ β© π΅)β©)) = π) | |
| 5 | 3, 4 | sylan9eqr 2788 | . 2 β’ ((π΅ β π΄ β§ (π β π β§ π΄ β π)) β π = π) |
| 6 | 5 | 3impb 1112 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β© cin 3942 β wss 3943 ifcif 4523 β¨cop 4629 βcfv 6536 (class class class)co 7404 sSet csts 17102 ndxcnx 17132 Basecbs 17150 βΎs cress 17179 |
| 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 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-ress 17180 |
| This theorem is referenced by: ressbas 17185 ressbasOLD 17186 resseqnbas 17192 resslemOLD 17193 ress0 17194 ressid 17195 ressinbas 17196 ressress 17199 rescabs 17788 rescabsOLD 17789 0symgefmndeq 19310 snsymgefmndeq 19311 mgpress 20051 mgpressOLD 20052 psgnghm2 21469 resvsca 32946 extdg1id 33259 |
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