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Mirrors > Home > MPE Home > Th. List > resslemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of resseqnbas 17195 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 2726 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
4 | 2, 3 | ressid2 17186 | . . . . . 6 β’ (((Baseβπ) β π΄ β§ π β V β§ π΄ β π) β π = π) |
5 | 4 | fveq2d 6889 | . . . . 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 17187 | . . . . . . 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 6889 | . . . . . 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 17139 | . . . . . . 7 β’ πΈ = Slot (πΈβndx) |
12 | 9, 10 | ndxarg 17138 | . . . . . . . . 9 β’ (πΈβndx) = π |
13 | 1re 11218 | . . . . . . . . . 10 β’ 1 β β | |
14 | resslemOLD.b | . . . . . . . . . 10 β’ 1 < π | |
15 | 13, 14 | gtneii 11330 | . . . . . . . . 9 β’ π β 1 |
16 | 12, 15 | eqnetri 3005 | . . . . . . . 8 β’ (πΈβndx) β 1 |
17 | basendx 17162 | . . . . . . . 8 β’ (Baseβndx) = 1 | |
18 | 16, 17 | neeqtrri 3008 | . . . . . . 7 β’ (πΈβndx) β (Baseβndx) |
19 | 11, 18 | setsnid 17151 | . . . . . 6 β’ (πΈβπ) = (πΈβ(π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) |
20 | 8, 19 | eqtr4di 2784 | . . . . 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 17184 | . . . . . . . . 9 β’ Rel dom βΎs | |
24 | 23 | ovprc1 7444 | . . . . . . . 8 β’ (Β¬ π β V β (π βΎs π΄) = β ) |
25 | 2, 24 | eqtrid 2778 | . . . . . . 7 β’ (Β¬ π β V β π = β ) |
26 | 25 | fveq2d 6889 | . . . . . 6 β’ (Β¬ π β V β (πΈβπ ) = (πΈββ )) |
27 | 9 | str0 17131 | . . . . . 6 β’ β = (πΈββ ) |
28 | 26, 27 | eqtr4di 2784 | . . . . 5 β’ (Β¬ π β V β (πΈβπ ) = β ) |
29 | fvprc 6877 | . . . . 5 β’ (Β¬ π β V β (πΈβπ) = β ) | |
30 | 28, 29 | eqtr4d 2769 | . . . 4 β’ (Β¬ π β V β (πΈβπ ) = (πΈβπ)) |
31 | 30 | adantr 480 | . . 3 β’ ((Β¬ π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ)) |
32 | 22, 31 | pm2.61ian 809 | . 2 β’ (π΄ β π β (πΈβπ ) = (πΈβπ)) |
33 | 1, 32 | eqtr4id 2785 | 1 β’ (π΄ β π β πΆ = (πΈβπ )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3468 β© cin 3942 β wss 3943 β c0 4317 β¨cop 4629 class class class wbr 5141 βcfv 6537 (class class class)co 7405 1c1 11113 < clt 11252 βcn 12216 sSet csts 17105 Slot cslot 17123 ndxcnx 17135 Basecbs 17153 βΎs cress 17182 |
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-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-mulcl 11174 ax-mulrcl 11175 ax-i2m1 11180 ax-1ne0 11181 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-nn 12217 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 |
This theorem is referenced by: (None) |
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