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Mirrors > Home > MPE Home > Th. List > resseqnbas | Structured version Visualization version GIF version |
Description: The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.) |
Ref | Expression |
---|---|
resseqnbas.r | β’ π = (π βΎs π΄) |
resseqnbas.e | β’ πΆ = (πΈβπ) |
resseqnbas.f | β’ πΈ = Slot (πΈβndx) |
resseqnbas.n | β’ (πΈβndx) β (Baseβndx) |
Ref | Expression |
---|---|
resseqnbas | β’ (π΄ β π β πΆ = (πΈβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resseqnbas.e | . 2 β’ πΆ = (πΈβπ) | |
2 | resseqnbas.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 | resseqnbas.f | . . . . . . 7 β’ πΈ = Slot (πΈβndx) | |
10 | resseqnbas.n | . . . . . . 7 β’ (πΈβndx) β (Baseβndx) | |
11 | 9, 10 | setsnid 17086 | . . . . . 6 β’ (πΈβπ) = (πΈβ(π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) |
12 | 8, 11 | eqtr4di 2791 | . . . . 5 β’ ((Β¬ (Baseβπ) β π΄ β§ π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ)) |
13 | 12 | 3expib 1123 | . . . 4 β’ (Β¬ (Baseβπ) β π΄ β ((π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ))) |
14 | 6, 13 | pm2.61i 182 | . . 3 β’ ((π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ)) |
15 | 9 | str0 17066 | . . . . . . 7 β’ β = (πΈββ ) |
16 | 15 | eqcomi 2742 | . . . . . 6 β’ (πΈββ ) = β |
17 | reldmress 17119 | . . . . . 6 β’ Rel dom βΎs | |
18 | 16, 2, 17 | oveqprc 17069 | . . . . 5 β’ (Β¬ π β V β (πΈβπ) = (πΈβπ )) |
19 | 18 | eqcomd 2739 | . . . 4 β’ (Β¬ π β V β (πΈβπ ) = (πΈβπ)) |
20 | 19 | adantr 482 | . . 3 β’ ((Β¬ π β V β§ π΄ β π) β (πΈβπ ) = (πΈβπ)) |
21 | 14, 20 | pm2.61ian 811 | . 2 β’ (π΄ β π β (πΈβπ ) = (πΈβπ)) |
22 | 1, 21 | 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 β wne 2940 Vcvv 3444 β© cin 3910 β wss 3911 β c0 4283 β¨cop 4593 βcfv 6497 (class class class)co 7358 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-pr 5385 ax-un 7673 |
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-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-res 5646 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-sets 17041 df-slot 17059 df-ress 17118 |
This theorem is referenced by: ressplusg 17176 ressmulr 17193 ressstarv 17194 resssca 17229 ressvsca 17230 ressip 17231 resstset 17251 ressle 17266 ressunif 17288 ressds 17296 resshom 17305 ressco 17306 |
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