![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > strslfv2d | GIF version |
Description: Deduction version of strslfv 12509. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.) |
Ref | Expression |
---|---|
strslfv2d.e | β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) |
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 |
---|---|
strslfv2d | β’ (π β πΆ = (πΈβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strslfv2d.e | . . . 4 β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) | |
2 | 1 | simpli 111 | . . 3 β’ πΈ = Slot (πΈβndx) |
3 | strfv2d.s | . . 3 β’ (π β π β π) | |
4 | 1 | simpri 113 | . . . 4 β’ (πΈβndx) β β |
5 | 4 | a1i 9 | . . 3 β’ (π β (πΈβndx) β β) |
6 | 2, 3, 5 | strnfvnd 12484 | . 2 β’ (π β (πΈβπ) = (πβ(πΈβndx))) |
7 | cnvcnv2 5084 | . . . 4 β’ β‘β‘π = (π βΎ V) | |
8 | 7 | fveq1i 5518 | . . 3 β’ (β‘β‘πβ(πΈβndx)) = ((π βΎ V)β(πΈβndx)) |
9 | 5 | elexd 2752 | . . . 4 β’ (π β (πΈβndx) β V) |
10 | fvres 5541 | . . . 4 β’ ((πΈβndx) β V β ((π βΎ V)β(πΈβndx)) = (πβ(πΈβndx))) | |
11 | 9, 10 | syl 14 | . . 3 β’ (π β ((π βΎ V)β(πΈβndx)) = (πβ(πΈβndx))) |
12 | 8, 11 | eqtrid 2222 | . 2 β’ (π β (β‘β‘πβ(πΈβndx)) = (πβ(πΈβndx))) |
13 | strfv2d.f | . . 3 β’ (π β Fun β‘β‘π) | |
14 | strfv2d.n | . . . . 5 β’ (π β β¨(πΈβndx), πΆβ© β π) | |
15 | strfv2d.c | . . . . . . 7 β’ (π β πΆ β π) | |
16 | 15 | elexd 2752 | . . . . . 6 β’ (π β πΆ β V) |
17 | 9, 16 | opelxpd 4661 | . . . . 5 β’ (π β β¨(πΈβndx), πΆβ© β (V Γ V)) |
18 | 14, 17 | elind 3322 | . . . 4 β’ (π β β¨(πΈβndx), πΆβ© β (π β© (V Γ V))) |
19 | cnvcnv 5083 | . . . 4 β’ β‘β‘π = (π β© (V Γ V)) | |
20 | 18, 19 | eleqtrrdi 2271 | . . 3 β’ (π β β¨(πΈβndx), πΆβ© β β‘β‘π) |
21 | funopfv 5557 | . . 3 β’ (Fun β‘β‘π β (β¨(πΈβndx), πΆβ© β β‘β‘π β (β‘β‘πβ(πΈβndx)) = πΆ)) | |
22 | 13, 20, 21 | sylc 62 | . 2 β’ (π β (β‘β‘πβ(πΈβndx)) = πΆ) |
23 | 6, 12, 22 | 3eqtr2rd 2217 | 1 β’ (π β πΆ = (πΈβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 Vcvv 2739 β© cin 3130 β¨cop 3597 Γ cxp 4626 β‘ccnv 4627 βΎ cres 4630 Fun wfun 5212 βcfv 5218 βcn 8921 ndxcnx 12461 Slot cslot 12463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-iota 5180 df-fun 5220 df-fv 5226 df-slot 12468 |
This theorem is referenced by: strslfv2 12508 strslfv 12509 opelstrsl 12575 |
Copyright terms: Public domain | W3C validator |