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Mirrors > Home > MPE Home > Th. List > usgrstrrepe | Structured version Visualization version GIF version |
Description: Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a simple graph. Instead of requiring (π β πΊ Struct π), it would be sufficient to require (π β Fun (πΊ β {β })) and (π β πΊ β V). (Contributed by AV, 13-Nov-2021.) (Proof shortened by AV, 16-Nov-2021.) |
Ref | Expression |
---|---|
usgrstrrepe.v | β’ π = (BaseβπΊ) |
usgrstrrepe.i | β’ πΌ = (.efβndx) |
usgrstrrepe.s | β’ (π β πΊ Struct π) |
usgrstrrepe.b | β’ (π β (Baseβndx) β dom πΊ) |
usgrstrrepe.w | β’ (π β πΈ β π) |
usgrstrrepe.e | β’ (π β πΈ:dom πΈβ1-1β{π₯ β π« π β£ (β―βπ₯) = 2}) |
Ref | Expression |
---|---|
usgrstrrepe | β’ (π β (πΊ sSet β¨πΌ, πΈβ©) β USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrstrrepe.e | . . . 4 β’ (π β πΈ:dom πΈβ1-1β{π₯ β π« π β£ (β―βπ₯) = 2}) | |
2 | usgrstrrepe.i | . . . . . . . . 9 β’ πΌ = (.efβndx) | |
3 | usgrstrrepe.s | . . . . . . . . 9 β’ (π β πΊ Struct π) | |
4 | usgrstrrepe.b | . . . . . . . . 9 β’ (π β (Baseβndx) β dom πΊ) | |
5 | usgrstrrepe.w | . . . . . . . . 9 β’ (π β πΈ β π) | |
6 | 2, 3, 4, 5 | setsvtx 28028 | . . . . . . . 8 β’ (π β (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) = (BaseβπΊ)) |
7 | usgrstrrepe.v | . . . . . . . 8 β’ π = (BaseβπΊ) | |
8 | 6, 7 | eqtr4di 2791 | . . . . . . 7 β’ (π β (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) = π) |
9 | 8 | pweqd 4578 | . . . . . 6 β’ (π β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) = π« π) |
10 | 9 | rabeqdv 3421 | . . . . 5 β’ (π β {π₯ β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) β£ (β―βπ₯) = 2} = {π₯ β π« π β£ (β―βπ₯) = 2}) |
11 | f1eq3 6736 | . . . . 5 β’ ({π₯ β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) β£ (β―βπ₯) = 2} = {π₯ β π« π β£ (β―βπ₯) = 2} β (πΈ:dom πΈβ1-1β{π₯ β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) β£ (β―βπ₯) = 2} β πΈ:dom πΈβ1-1β{π₯ β π« π β£ (β―βπ₯) = 2})) | |
12 | 10, 11 | syl 17 | . . . 4 β’ (π β (πΈ:dom πΈβ1-1β{π₯ β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) β£ (β―βπ₯) = 2} β πΈ:dom πΈβ1-1β{π₯ β π« π β£ (β―βπ₯) = 2})) |
13 | 1, 12 | mpbird 257 | . . 3 β’ (π β πΈ:dom πΈβ1-1β{π₯ β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) β£ (β―βπ₯) = 2}) |
14 | 2, 3, 4, 5 | setsiedg 28029 | . . . 4 β’ (π β (iEdgβ(πΊ sSet β¨πΌ, πΈβ©)) = πΈ) |
15 | 14 | dmeqd 5862 | . . . 4 β’ (π β dom (iEdgβ(πΊ sSet β¨πΌ, πΈβ©)) = dom πΈ) |
16 | eqidd 2734 | . . . 4 β’ (π β {π₯ β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) β£ (β―βπ₯) = 2} = {π₯ β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) β£ (β―βπ₯) = 2}) | |
17 | 14, 15, 16 | f1eq123d 6777 | . . 3 β’ (π β ((iEdgβ(πΊ sSet β¨πΌ, πΈβ©)):dom (iEdgβ(πΊ sSet β¨πΌ, πΈβ©))β1-1β{π₯ β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) β£ (β―βπ₯) = 2} β πΈ:dom πΈβ1-1β{π₯ β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) β£ (β―βπ₯) = 2})) |
18 | 13, 17 | mpbird 257 | . 2 β’ (π β (iEdgβ(πΊ sSet β¨πΌ, πΈβ©)):dom (iEdgβ(πΊ sSet β¨πΌ, πΈβ©))β1-1β{π₯ β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) β£ (β―βπ₯) = 2}) |
19 | ovex 7391 | . . 3 β’ (πΊ sSet β¨πΌ, πΈβ©) β V | |
20 | eqid 2733 | . . . 4 β’ (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) = (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) | |
21 | eqid 2733 | . . . 4 β’ (iEdgβ(πΊ sSet β¨πΌ, πΈβ©)) = (iEdgβ(πΊ sSet β¨πΌ, πΈβ©)) | |
22 | 20, 21 | isusgrs 28149 | . . 3 β’ ((πΊ sSet β¨πΌ, πΈβ©) β V β ((πΊ sSet β¨πΌ, πΈβ©) β USGraph β (iEdgβ(πΊ sSet β¨πΌ, πΈβ©)):dom (iEdgβ(πΊ sSet β¨πΌ, πΈβ©))β1-1β{π₯ β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) β£ (β―βπ₯) = 2})) |
23 | 19, 22 | mp1i 13 | . 2 β’ (π β ((πΊ sSet β¨πΌ, πΈβ©) β USGraph β (iEdgβ(πΊ sSet β¨πΌ, πΈβ©)):dom (iEdgβ(πΊ sSet β¨πΌ, πΈβ©))β1-1β{π₯ β π« (Vtxβ(πΊ sSet β¨πΌ, πΈβ©)) β£ (β―βπ₯) = 2})) |
24 | 18, 23 | mpbird 257 | 1 β’ (π β (πΊ sSet β¨πΌ, πΈβ©) β USGraph) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 {crab 3406 Vcvv 3444 π« cpw 4561 β¨cop 4593 class class class wbr 5106 dom cdm 5634 β1-1βwf1 6494 βcfv 6497 (class class class)co 7358 2c2 12213 β―chash 14236 Struct cstr 17023 sSet csts 17040 ndxcnx 17070 Basecbs 17088 .efcedgf 27979 Vtxcvtx 27989 iEdgciedg 27990 USGraphcusgr 28142 |
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-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-oadd 8417 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-dju 9842 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-xnn0 12491 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-hash 14237 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-edgf 27980 df-vtx 27991 df-iedg 27992 df-usgr 28144 |
This theorem is referenced by: structtousgr 28435 |
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