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Mirrors > Home > MPE Home > Th. List > resslemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of resseqnbas 17127 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 2733 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
4 | 2, 3 | ressid2 17121 | . . . . . 6 β’ (((Baseβπ) β π΄ β§ π β V β§ π΄ β π) β π = π) |
5 | 4 | fveq2d 6847 | . . . . 5 β’ (((Baseβπ) β π΄ β§ π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ)) |
6 | 5 | 3expib 1123 | . . . 4 β’ ((Baseβπ) β π΄ β ((π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ))) |
7 | 2, 3 | ressval2 17122 | . . . . . . 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 6847 | . . . . . 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 17074 | . . . . . . 7 β’ πΈ = Slot (πΈβndx) |
12 | 9, 10 | ndxarg 17073 | . . . . . . . . 9 β’ (πΈβndx) = π |
13 | 1re 11160 | . . . . . . . . . 10 β’ 1 β β | |
14 | resslemOLD.b | . . . . . . . . . 10 β’ 1 < π | |
15 | 13, 14 | gtneii 11272 | . . . . . . . . 9 β’ π β 1 |
16 | 12, 15 | eqnetri 3011 | . . . . . . . 8 β’ (πΈβndx) β 1 |
17 | basendx 17097 | . . . . . . . 8 β’ (Baseβndx) = 1 | |
18 | 16, 17 | neeqtrri 3014 | . . . . . . 7 β’ (πΈβndx) β (Baseβndx) |
19 | 11, 18 | setsnid 17086 | . . . . . 6 β’ (πΈβπ) = (πΈβ(π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) |
20 | 8, 19 | eqtr4di 2791 | . . . . 5 β’ ((Β¬ (Baseβπ) β π΄ β§ π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ)) |
21 | 20 | 3expib 1123 | . . . 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 17119 | . . . . . . . . 9 β’ Rel dom βΎs | |
24 | 23 | ovprc1 7397 | . . . . . . . 8 β’ (Β¬ π β V β (π βΎs π΄) = β ) |
25 | 2, 24 | eqtrid 2785 | . . . . . . 7 β’ (Β¬ π β V β π = β ) |
26 | 25 | fveq2d 6847 | . . . . . 6 β’ (Β¬ π β V β (πΈβπ ) = (πΈββ )) |
27 | 9 | str0 17066 | . . . . . 6 β’ β = (πΈββ ) |
28 | 26, 27 | eqtr4di 2791 | . . . . 5 β’ (Β¬ π β V β (πΈβπ ) = β ) |
29 | fvprc 6835 | . . . . 5 β’ (Β¬ π β V β (πΈβπ) = β ) | |
30 | 28, 29 | eqtr4d 2776 | . . . 4 β’ (Β¬ π β V β (πΈβπ ) = (πΈβπ)) |
31 | 30 | adantr 482 | . . 3 β’ ((Β¬ π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ)) |
32 | 22, 31 | pm2.61ian 811 | . 2 β’ (π΄ β π β (πΈβπ ) = (πΈβπ)) |
33 | 1, 32 | eqtr4id 2792 | 1 β’ (π΄ β π β πΆ = (πΈβπ )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3444 β© cin 3910 β wss 3911 β c0 4283 β¨cop 4593 class class class wbr 5106 βcfv 6497 (class class class)co 7358 1c1 11057 < clt 11194 βcn 12158 sSet csts 17040 Slot cslot 17058 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-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-mulcl 11118 ax-mulrcl 11119 ax-i2m1 11124 ax-1ne0 11125 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-ltxr 11199 df-nn 12159 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 |
This theorem is referenced by: (None) |
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