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Mirrors > Home > MPE Home > Th. List > strfvd | Structured version Visualization version GIF version |
Description: Deduction version of strfv 16950. (Contributed by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
strfvd.e | β’ πΈ = Slot (πΈβndx) |
strfvd.s | β’ (π β π β π) |
strfvd.f | β’ (π β Fun π) |
strfvd.n | β’ (π β β¨(πΈβndx), πΆβ© β π) |
Ref | Expression |
---|---|
strfvd | β’ (π β πΆ = (πΈβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strfvd.e | . . 3 β’ πΈ = Slot (πΈβndx) | |
2 | strfvd.s | . . 3 β’ (π β π β π) | |
3 | 1, 2 | strfvnd 16931 | . 2 β’ (π β (πΈβπ) = (πβ(πΈβndx))) |
4 | strfvd.f | . . 3 β’ (π β Fun π) | |
5 | strfvd.n | . . 3 β’ (π β β¨(πΈβndx), πΆβ© β π) | |
6 | funopfv 6853 | . . 3 β’ (Fun π β (β¨(πΈβndx), πΆβ© β π β (πβ(πΈβndx)) = πΆ)) | |
7 | 4, 5, 6 | sylc 65 | . 2 β’ (π β (πβ(πΈβndx)) = πΆ) |
8 | 3, 7 | eqtr2d 2777 | 1 β’ (π β πΆ = (πΈβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 β¨cop 4571 Fun wfun 6452 βcfv 6458 Slot cslot 16927 ndxcnx 16939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-iota 6410 df-fun 6460 df-fv 6466 df-slot 16928 |
This theorem is referenced by: strssd 16952 |
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