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Mirrors > Home > MPE Home > Th. List > edgfiedgval | Structured version Visualization version GIF version |
Description: The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.) |
Ref | Expression |
---|---|
basvtxval.s | β’ (π β πΊ Struct π) |
basvtxval.d | β’ (π β 2 β€ (β―βdom πΊ)) |
edgfiedgval.e | β’ (π β πΈ β π) |
edgfiedgval.f | β’ (π β β¨(.efβndx), πΈβ© β πΊ) |
Ref | Expression |
---|---|
edgfiedgval | β’ (π β (iEdgβπΊ) = πΈ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | basvtxval.s | . . . 4 β’ (π β πΊ Struct π) | |
2 | structn0fun 16959 | . . . 4 β’ (πΊ Struct π β Fun (πΊ β {β })) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π β Fun (πΊ β {β })) |
4 | basvtxval.d | . . 3 β’ (π β 2 β€ (β―βdom πΊ)) | |
5 | funiedgdmge2val 27768 | . . 3 β’ ((Fun (πΊ β {β }) β§ 2 β€ (β―βdom πΊ)) β (iEdgβπΊ) = (.efβπΊ)) | |
6 | 3, 4, 5 | syl2anc 585 | . 2 β’ (π β (iEdgβπΊ) = (.efβπΊ)) |
7 | edgfid 27744 | . . 3 β’ .ef = Slot (.efβndx) | |
8 | structex 16958 | . . . 4 β’ (πΊ Struct π β πΊ β V) | |
9 | 1, 8 | syl 17 | . . 3 β’ (π β πΊ β V) |
10 | structfung 16962 | . . . 4 β’ (πΊ Struct π β Fun β‘β‘πΊ) | |
11 | 1, 10 | syl 17 | . . 3 β’ (π β Fun β‘β‘πΊ) |
12 | edgfiedgval.f | . . 3 β’ (π β β¨(.efβndx), πΈβ© β πΊ) | |
13 | edgfiedgval.e | . . 3 β’ (π β πΈ β π) | |
14 | 7, 9, 11, 12, 13 | strfv2d 17010 | . 2 β’ (π β πΈ = (.efβπΊ)) |
15 | 6, 14 | eqtr4d 2781 | 1 β’ (π β (iEdgβπΊ) = πΈ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3444 β cdif 3906 β c0 4281 {csn 4585 β¨cop 4591 class class class wbr 5104 β‘ccnv 5630 dom cdm 5631 Fun wfun 6486 βcfv 6492 β€ cle 11124 2c2 12142 β―chash 14159 Struct cstr 16954 ndxcnx 17001 .efcedgf 27742 iEdgciedg 27753 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-xnn0 12420 df-z 12434 df-dec 12553 df-uz 12698 df-fz 13355 df-hash 14160 df-struct 16955 df-slot 16990 df-ndx 17002 df-edgf 27743 df-iedg 27755 |
This theorem is referenced by: structgrssiedg 27781 |
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