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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 17219 | . . 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 4538 | . . 3 β’ (π΅ β π΄ β if(π΅ β π΄, π, (π sSet β¨(Baseβndx), (π΄ β© π΅)β©)) = π) | |
| 5 | 3, 4 | sylan9eqr 2790 | . 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β© cin 3948 β wss 3949 ifcif 4532 β¨cop 4638 βcfv 6553 (class class class)co 7426 sSet csts 17139 ndxcnx 17169 Basecbs 17187 βΎs cress 17216 |
| 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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-ress 17217 |
| This theorem is referenced by: ressbas 17222 ressbasOLD 17223 resseqnbas 17229 resslemOLD 17230 ress0 17231 ressid 17232 ressinbas 17233 ressress 17236 rescabs 17825 rescabsOLD 17826 0symgefmndeq 19355 snsymgefmndeq 19356 mgpress 20096 mgpressOLD 20097 psgnghm2 21520 resvsca 33065 extdg1id 33388 |
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