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Mirrors > Home > MPE Home > Th. List > strfv2d | Structured version Visualization version GIF version |
Description: Deduction version of strfv2 17080. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
strfv2d.e | β’ πΈ = Slot (πΈβndx) |
strfv2d.s | β’ (π β π β π) |
strfv2d.f | β’ (π β Fun β‘β‘π) |
strfv2d.n | β’ (π β β¨(πΈβndx), πΆβ© β π) |
strfv2d.c | β’ (π β πΆ β π) |
Ref | Expression |
---|---|
strfv2d | β’ (π β πΆ = (πΈβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strfv2d.e | . . 3 β’ πΈ = Slot (πΈβndx) | |
2 | strfv2d.s | . . 3 β’ (π β π β π) | |
3 | 1, 2 | strfvnd 17062 | . 2 β’ (π β (πΈβπ) = (πβ(πΈβndx))) |
4 | cnvcnv2 6146 | . . . . 5 β’ β‘β‘π = (π βΎ V) | |
5 | 4 | fveq1i 6844 | . . . 4 β’ (β‘β‘πβ(πΈβndx)) = ((π βΎ V)β(πΈβndx)) |
6 | fvex 6856 | . . . . 5 β’ (πΈβndx) β V | |
7 | fvres 6862 | . . . . 5 β’ ((πΈβndx) β V β ((π βΎ V)β(πΈβndx)) = (πβ(πΈβndx))) | |
8 | 6, 7 | ax-mp 5 | . . . 4 β’ ((π βΎ V)β(πΈβndx)) = (πβ(πΈβndx)) |
9 | 5, 8 | eqtri 2761 | . . 3 β’ (β‘β‘πβ(πΈβndx)) = (πβ(πΈβndx)) |
10 | strfv2d.f | . . . 4 β’ (π β Fun β‘β‘π) | |
11 | strfv2d.n | . . . . . 6 β’ (π β β¨(πΈβndx), πΆβ© β π) | |
12 | strfv2d.c | . . . . . . . 8 β’ (π β πΆ β π) | |
13 | 12 | elexd 3464 | . . . . . . 7 β’ (π β πΆ β V) |
14 | opelxpi 5671 | . . . . . . 7 β’ (((πΈβndx) β V β§ πΆ β V) β β¨(πΈβndx), πΆβ© β (V Γ V)) | |
15 | 6, 13, 14 | sylancr 588 | . . . . . 6 β’ (π β β¨(πΈβndx), πΆβ© β (V Γ V)) |
16 | 11, 15 | elind 4155 | . . . . 5 β’ (π β β¨(πΈβndx), πΆβ© β (π β© (V Γ V))) |
17 | cnvcnv 6145 | . . . . 5 β’ β‘β‘π = (π β© (V Γ V)) | |
18 | 16, 17 | eleqtrrdi 2845 | . . . 4 β’ (π β β¨(πΈβndx), πΆβ© β β‘β‘π) |
19 | funopfv 6895 | . . . 4 β’ (Fun β‘β‘π β (β¨(πΈβndx), πΆβ© β β‘β‘π β (β‘β‘πβ(πΈβndx)) = πΆ)) | |
20 | 10, 18, 19 | sylc 65 | . . 3 β’ (π β (β‘β‘πβ(πΈβndx)) = πΆ) |
21 | 9, 20 | eqtr3id 2787 | . 2 β’ (π β (πβ(πΈβndx)) = πΆ) |
22 | 3, 21 | eqtr2d 2774 | 1 β’ (π β πΆ = (πΈβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3444 β© cin 3910 β¨cop 4593 Γ cxp 5632 β‘ccnv 5633 βΎ cres 5636 Fun wfun 6491 βcfv 6497 Slot cslot 17058 ndxcnx 17070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-res 5646 df-iota 6449 df-fun 6499 df-fv 6505 df-slot 17059 |
This theorem is referenced by: strfv2 17080 opelstrbas 17102 ebtwntg 27973 ecgrtg 27974 elntg 27975 edgfiedgval 28010 |
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