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Mirrors > Home > MPE Home > Th. List > resslemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of resseqnbas 17219 as of 21-Oct-2024. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
resslemOLD.r | β’ π = (π βΎs π΄) |
resslemOLD.e | β’ πΆ = (πΈβπ) |
resslemOLD.f | β’ πΈ = Slot π |
resslemOLD.n | β’ π β β |
resslemOLD.b | β’ 1 < π |
Ref | Expression |
---|---|
resslemOLD | β’ (π΄ β π β πΆ = (πΈβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resslemOLD.e | . 2 β’ πΆ = (πΈβπ) | |
2 | resslemOLD.r | . . . . . . 7 β’ π = (π βΎs π΄) | |
3 | eqid 2725 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
4 | 2, 3 | ressid2 17210 | . . . . . 6 β’ (((Baseβπ) β π΄ β§ π β V β§ π΄ β π) β π = π) |
5 | 4 | fveq2d 6895 | . . . . 5 β’ (((Baseβπ) β π΄ β§ π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ)) |
6 | 5 | 3expib 1119 | . . . 4 β’ ((Baseβπ) β π΄ β ((π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ))) |
7 | 2, 3 | ressval2 17211 | . . . . . . 7 β’ ((Β¬ (Baseβπ) β π΄ β§ π β V β§ π΄ β π) β π = (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) |
8 | 7 | fveq2d 6895 | . . . . . 6 β’ ((Β¬ (Baseβπ) β π΄ β§ π β V β§ π΄ β π) β (πΈβπ ) = (πΈβ(π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©))) |
9 | resslemOLD.f | . . . . . . . 8 β’ πΈ = Slot π | |
10 | resslemOLD.n | . . . . . . . 8 β’ π β β | |
11 | 9, 10 | ndxid 17163 | . . . . . . 7 β’ πΈ = Slot (πΈβndx) |
12 | 9, 10 | ndxarg 17162 | . . . . . . . . 9 β’ (πΈβndx) = π |
13 | 1re 11242 | . . . . . . . . . 10 β’ 1 β β | |
14 | resslemOLD.b | . . . . . . . . . 10 β’ 1 < π | |
15 | 13, 14 | gtneii 11354 | . . . . . . . . 9 β’ π β 1 |
16 | 12, 15 | eqnetri 3001 | . . . . . . . 8 β’ (πΈβndx) β 1 |
17 | basendx 17186 | . . . . . . . 8 β’ (Baseβndx) = 1 | |
18 | 16, 17 | neeqtrri 3004 | . . . . . . 7 β’ (πΈβndx) β (Baseβndx) |
19 | 11, 18 | setsnid 17175 | . . . . . 6 β’ (πΈβπ) = (πΈβ(π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) |
20 | 8, 19 | eqtr4di 2783 | . . . . 5 β’ ((Β¬ (Baseβπ) β π΄ β§ π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ)) |
21 | 20 | 3expib 1119 | . . . 4 β’ (Β¬ (Baseβπ) β π΄ β ((π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ))) |
22 | 6, 21 | pm2.61i 182 | . . 3 β’ ((π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ)) |
23 | reldmress 17208 | . . . . . . . . 9 β’ Rel dom βΎs | |
24 | 23 | ovprc1 7454 | . . . . . . . 8 β’ (Β¬ π β V β (π βΎs π΄) = β ) |
25 | 2, 24 | eqtrid 2777 | . . . . . . 7 β’ (Β¬ π β V β π = β ) |
26 | 25 | fveq2d 6895 | . . . . . 6 β’ (Β¬ π β V β (πΈβπ ) = (πΈββ )) |
27 | 9 | str0 17155 | . . . . . 6 β’ β = (πΈββ ) |
28 | 26, 27 | eqtr4di 2783 | . . . . 5 β’ (Β¬ π β V β (πΈβπ ) = β ) |
29 | fvprc 6883 | . . . . 5 β’ (Β¬ π β V β (πΈβπ) = β ) | |
30 | 28, 29 | eqtr4d 2768 | . . . 4 β’ (Β¬ π β V β (πΈβπ ) = (πΈβπ)) |
31 | 30 | adantr 479 | . . 3 β’ ((Β¬ π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ)) |
32 | 22, 31 | pm2.61ian 810 | . 2 β’ (π΄ β π β (πΈβπ ) = (πΈβπ)) |
33 | 1, 32 | eqtr4id 2784 | 1 β’ (π΄ β π β πΆ = (πΈβπ )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3463 β© cin 3939 β wss 3940 β c0 4318 β¨cop 4630 class class class wbr 5143 βcfv 6542 (class class class)co 7415 1c1 11137 < clt 11276 βcn 12240 sSet csts 17129 Slot cslot 17147 ndxcnx 17159 Basecbs 17177 βΎs cress 17206 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-mulcl 11198 ax-mulrcl 11199 ax-i2m1 11204 ax-1ne0 11205 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-ltxr 11281 df-nn 12241 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 |
This theorem is referenced by: (None) |
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