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Mirrors > Home > ILE Home > Th. List > strslfv2d | GIF version |
Description: Deduction version of strslfv 12521. (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 12496 | . 2 β’ (π β (πΈβπ) = (πβ(πΈβndx))) |
7 | cnvcnv2 5094 | . . . 4 β’ β‘β‘π = (π βΎ V) | |
8 | 7 | fveq1i 5528 | . . 3 β’ (β‘β‘πβ(πΈβndx)) = ((π βΎ V)β(πΈβndx)) |
9 | 5 | elexd 2762 | . . . 4 β’ (π β (πΈβndx) β V) |
10 | fvres 5551 | . . . 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 2232 | . 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 2762 | . . . . . 6 β’ (π β πΆ β V) |
17 | 9, 16 | opelxpd 4671 | . . . . 5 β’ (π β β¨(πΈβndx), πΆβ© β (V Γ V)) |
18 | 14, 17 | elind 3332 | . . . 4 β’ (π β β¨(πΈβndx), πΆβ© β (π β© (V Γ V))) |
19 | cnvcnv 5093 | . . . 4 β’ β‘β‘π = (π β© (V Γ V)) | |
20 | 18, 19 | eleqtrrdi 2281 | . . 3 β’ (π β β¨(πΈβndx), πΆβ© β β‘β‘π) |
21 | funopfv 5568 | . . 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 2227 | 1 β’ (π β πΆ = (πΈβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1363 β wcel 2158 Vcvv 2749 β© cin 3140 β¨cop 3607 Γ cxp 4636 β‘ccnv 4637 βΎ cres 4640 Fun wfun 5222 βcfv 5228 βcn 8933 ndxcnx 12473 Slot cslot 12475 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-iota 5190 df-fun 5230 df-fv 5236 df-slot 12480 |
This theorem is referenced by: strslfv2 12520 strslfv 12521 opelstrsl 12588 |
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