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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 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 | iftrue 4493 | . . 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 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 ress0 17129 ressid 17130 ressinbas 17131 ressress 17134 rescabs 17723 rescabsOLD 17724 0symgefmndeq 19180 snsymgefmndeq 19181 mgpress 19916 mgpressOLD 19917 psgnghm2 21001 resvsca 32168 extdg1id 32409 |
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