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Mirrors > Home > MPE Home > Th. List > strfv2d | Structured version Visualization version GIF version |
Description: Deduction version of strfv2 17135. (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 17117 | . 2 β’ (π β (πΈβπ) = (πβ(πΈβndx))) |
4 | cnvcnv2 6192 | . . . . 5 β’ β‘β‘π = (π βΎ V) | |
5 | 4 | fveq1i 6892 | . . . 4 β’ (β‘β‘πβ(πΈβndx)) = ((π βΎ V)β(πΈβndx)) |
6 | fvex 6904 | . . . . 5 β’ (πΈβndx) β V | |
7 | fvres 6910 | . . . . 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 2760 | . . 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 3494 | . . . . . . 7 β’ (π β πΆ β V) |
14 | opelxpi 5713 | . . . . . . 7 β’ (((πΈβndx) β V β§ πΆ β V) β β¨(πΈβndx), πΆβ© β (V Γ V)) | |
15 | 6, 13, 14 | sylancr 587 | . . . . . 6 β’ (π β β¨(πΈβndx), πΆβ© β (V Γ V)) |
16 | 11, 15 | elind 4194 | . . . . 5 β’ (π β β¨(πΈβndx), πΆβ© β (π β© (V Γ V))) |
17 | cnvcnv 6191 | . . . . 5 β’ β‘β‘π = (π β© (V Γ V)) | |
18 | 16, 17 | eleqtrrdi 2844 | . . . 4 β’ (π β β¨(πΈβndx), πΆβ© β β‘β‘π) |
19 | funopfv 6943 | . . . 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 2786 | . 2 β’ (π β (πβ(πΈβndx)) = πΆ) |
22 | 3, 21 | eqtr2d 2773 | 1 β’ (π β πΆ = (πΈβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 Vcvv 3474 β© cin 3947 β¨cop 4634 Γ cxp 5674 β‘ccnv 5675 βΎ cres 5678 Fun wfun 6537 βcfv 6543 Slot cslot 17113 ndxcnx 17125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-res 5688 df-iota 6495 df-fun 6545 df-fv 6551 df-slot 17114 |
This theorem is referenced by: strfv2 17135 opelstrbas 17157 ebtwntg 28237 ecgrtg 28238 elntg 28239 edgfiedgval 28274 |
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