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Mirrors > Home > MPE Home > Th. List > strss | Structured version Visualization version GIF version |
Description: Propagate component extraction to a structure π from a subset structure π. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 15-Jan-2014.) |
Ref | Expression |
---|---|
strss.t | β’ π β V |
strss.f | β’ Fun π |
strss.s | β’ π β π |
strss.e | β’ πΈ = Slot (πΈβndx) |
strss.n | β’ β¨(πΈβndx), πΆβ© β π |
Ref | Expression |
---|---|
strss | β’ (πΈβπ) = (πΈβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strss.e | . . 3 β’ πΈ = Slot (πΈβndx) | |
2 | strss.t | . . . 4 β’ π β V | |
3 | 2 | a1i 11 | . . 3 β’ (β€ β π β V) |
4 | strss.f | . . . 4 β’ Fun π | |
5 | 4 | a1i 11 | . . 3 β’ (β€ β Fun π) |
6 | strss.s | . . . 4 β’ π β π | |
7 | 6 | a1i 11 | . . 3 β’ (β€ β π β π) |
8 | strss.n | . . . 4 β’ β¨(πΈβndx), πΆβ© β π | |
9 | 8 | a1i 11 | . . 3 β’ (β€ β β¨(πΈβndx), πΆβ© β π) |
10 | 1, 3, 5, 7, 9 | strssd 17144 | . 2 β’ (β€ β (πΈβπ) = (πΈβπ)) |
11 | 10 | mptru 1540 | 1 β’ (πΈβπ) = (πΈβπ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β€wtru 1534 β wcel 2098 Vcvv 3466 β wss 3941 β¨cop 4627 Fun wfun 6528 βcfv 6534 Slot cslot 17119 ndxcnx 17131 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fv 6542 df-slot 17120 |
This theorem is referenced by: grpss 18880 |
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