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Mirrors > Home > MPE Home > Th. List > strfvd | Structured version Visualization version GIF version |
Description: Deduction version of strfv 17142. (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 17123 | . 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 6934 | . . 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 2765 | 1 β’ (π β πΆ = (πΈβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 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: strssd 17144 |
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