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Mirrors > Home > MPE Home > Th. List > strfvd | Structured version Visualization version GIF version |
Description: Deduction version of strfv 17173. (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 17154 | . 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 6949 | . . 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 2769 | 1 β’ (π β πΆ = (πΈβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β¨cop 4635 Fun wfun 6542 βcfv 6548 Slot cslot 17150 ndxcnx 17162 |
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 17151 |
This theorem is referenced by: strssd 17175 |
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