![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 17174 | . 2 β’ (β€ β (πΈβπ) = (πΈβπ)) |
11 | 10 | mptru 1541 | 1 β’ (πΈβπ) = (πΈβπ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 β€wtru 1535 β wcel 2099 Vcvv 3471 β wss 3947 β¨cop 4635 Fun wfun 6542 βcfv 6548 Slot cslot 17149 ndxcnx 17161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-slot 17150 |
This theorem is referenced by: grpss 18910 |
Copyright terms: Public domain | W3C validator |