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Mirrors > Home > MPE Home > Th. List > strssd | Structured version Visualization version GIF version |
Description: Deduction version of strss 17147. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
strssd.e | β’ πΈ = Slot (πΈβndx) |
strssd.t | β’ (π β π β π) |
strssd.f | β’ (π β Fun π) |
strssd.s | β’ (π β π β π) |
strssd.n | β’ (π β β¨(πΈβndx), πΆβ© β π) |
Ref | Expression |
---|---|
strssd | β’ (π β (πΈβπ) = (πΈβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strssd.e | . . 3 β’ πΈ = Slot (πΈβndx) | |
2 | strssd.t | . . 3 β’ (π β π β π) | |
3 | strssd.f | . . 3 β’ (π β Fun π) | |
4 | strssd.s | . . . 4 β’ (π β π β π) | |
5 | strssd.n | . . . 4 β’ (π β β¨(πΈβndx), πΆβ© β π) | |
6 | 4, 5 | sseldd 3978 | . . 3 β’ (π β β¨(πΈβndx), πΆβ© β π) |
7 | 1, 2, 3, 6 | strfvd 17141 | . 2 β’ (π β πΆ = (πΈβπ)) |
8 | 2, 4 | ssexd 5317 | . . 3 β’ (π β π β V) |
9 | funss 6560 | . . . 4 β’ (π β π β (Fun π β Fun π)) | |
10 | 4, 3, 9 | sylc 65 | . . 3 β’ (π β Fun π) |
11 | 1, 8, 10, 5 | strfvd 17141 | . 2 β’ (π β πΆ = (πΈβπ)) |
12 | 7, 11 | eqtr3d 2768 | 1 β’ (π β (πΈβπ) = (πΈβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 β¨cop 4629 Fun wfun 6530 βcfv 6536 Slot cslot 17121 ndxcnx 17133 |
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-pr 5420 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-slot 17122 |
This theorem is referenced by: strss 17147 |
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