Theorem edgstruct 16059

Description: The edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 13-Oct-2020.)

Hypothesis

Ref	Expression
edgstruct.s	|- G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }

Assertion

Ref	Expression
edgstruct	|- ( ( V e. W /\ E e. X ) -> (Edg ` G ) = ran E )

Proof of Theorem edgstruct

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-edg 16053	. . 3 |- Edg = ( g e. _V |-> ran (iEdg ` g ) )
2		fveq2 5670	. . . 4 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
3	2	rneqd 4986	. . 3 |- ( g = G -> ran (iEdg ` g ) = ran (iEdg ` G ) )
4		edgstruct.s	. . . 4 |- G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }
5		basendxnn 13268	. . . . . 6 |- ( Base ` ndx ) e. NN
6		simpl 109	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> V e. W )
7		opexg 4344	. . . . . 6 |- ( ( ( Base ` ndx ) e. NN /\ V e. W ) -> <. ( Base ` ndx ) , V >. e. _V )
8	5, 6, 7	sylancr 414	. . . . 5 |- ( ( V e. W /\ E e. X ) -> <. ( Base ` ndx ) , V >. e. _V )
9		edgfndxnn 16003	. . . . . 6 |- (.ef ` ndx ) e. NN
10		simpr 110	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> E e. X )
11		opexg 4344	. . . . . 6 |- ( ( (.ef ` ndx ) e. NN /\ E e. X ) -> <. (.ef ` ndx ) , E >. e. _V )
12	9, 10, 11	sylancr 414	. . . . 5 |- ( ( V e. W /\ E e. X ) -> <. (.ef ` ndx ) , E >. e. _V )
13		prexg 4325	. . . . 5 |- ( ( <. ( Base ` ndx ) , V >. e. _V /\ <. (.ef ` ndx ) , E >. e. _V ) -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } e. _V )
14	8, 12, 13	syl2anc 411	. . . 4 |- ( ( V e. W /\ E e. X ) -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } e. _V )
15	4, 14	eqeltrid 2319	. . 3 |- ( ( V e. W /\ E e. X ) -> G e. _V )
16	5	elexi 2826	. . . . . 6 |- ( Base ` ndx ) e. _V
17	9	elexi 2826	. . . . . 6 |- (.ef ` ndx ) e. _V
18	5	a1i 9	. . . . . . . . 9 |- ( ( V e. W /\ E e. X ) -> ( Base ` ndx ) e. NN )
19	9	a1i 9	. . . . . . . . 9 |- ( ( V e. W /\ E e. X ) -> (.ef ` ndx ) e. NN )
20		basendxnedgfndx 16006	. . . . . . . . . 10 |- ( Base ` ndx ) =/= (.ef ` ndx )
21	20	a1i 9	. . . . . . . . 9 |- ( ( V e. W /\ E e. X ) -> ( Base ` ndx ) =/= (.ef ` ndx ) )
22		fnprg 5411	. . . . . . . . 9 |- ( ( ( ( Base ` ndx ) e. NN /\ (.ef ` ndx ) e. NN ) /\ ( V e. W /\ E e. X ) /\ ( Base ` ndx ) =/= (.ef ` ndx ) ) -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } Fn { ( Base ` ndx ) , (.ef ` ndx ) } )
23	18, 19, 6, 10, 21, 22	syl221anc 1285	. . . . . . . 8 |- ( ( V e. W /\ E e. X ) -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } Fn { ( Base ` ndx ) , (.ef ` ndx ) } )
24	4	fneq1i 5450	. . . . . . . 8 |- ( G Fn { ( Base ` ndx ) , (.ef ` ndx ) } <-> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } Fn { ( Base ` ndx ) , (.ef ` ndx ) } )
25	23, 24	sylibr 134	. . . . . . 7 |- ( ( V e. W /\ E e. X ) -> G Fn { ( Base ` ndx ) , (.ef ` ndx ) } )
26		fnfun 5453	. . . . . . 7 |- ( G Fn { ( Base ` ndx ) , (.ef ` ndx ) } -> Fun G )
27		fundif 5400	. . . . . . 7 |- ( Fun G -> Fun ( G \ { (/) } ) )
28	25, 26, 27	3syl 17	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> Fun ( G \ { (/) } ) )
29	25	fndmd 5457	. . . . . . 7 |- ( ( V e. W /\ E e. X ) -> dom G = { ( Base ` ndx ) , (.ef ` ndx ) } )
30		eqimss2 3293	. . . . . . 7 |- ( dom G = { ( Base ` ndx ) , (.ef ` ndx ) } -> { ( Base ` ndx ) , (.ef ` ndx ) } C_ dom G )
31	29, 30	syl 14	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> { ( Base ` ndx ) , (.ef ` ndx ) } C_ dom G )
32	16, 17, 15, 28, 21, 31	funiedgdm2vald 16027	. . . . 5 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) = (.ef ` G ) )
33		edgfid 16001	. . . . . . . 8 |- .ef = Slot (.ef ` ndx )
34	33, 9	ndxslid 13237	. . . . . . 7 |- (.ef = Slot (.ef ` ndx ) /\ (.ef ` ndx ) e. NN )
35	34	slotex 13239	. . . . . 6 |- ( G e. _V -> (.ef ` G ) e. _V )
36	15, 35	syl 14	. . . . 5 |- ( ( V e. W /\ E e. X ) -> (.ef ` G ) e. _V )
37	32, 36	eqeltrd 2309	. . . 4 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) e. _V )
38		rnexg 5022	. . . 4 |- ( (iEdg ` G ) e. _V -> ran (iEdg ` G ) e. _V )
39	37, 38	syl 14	. . 3 |- ( ( V e. W /\ E e. X ) -> ran (iEdg ` G ) e. _V )
40	1, 3, 15, 39	fvmptd3 5771	. 2 |- ( ( V e. W /\ E e. X ) -> (Edg ` G ) = ran (iEdg ` G ) )
41	4	struct2griedg 16041	. . 3 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) = E )
42	41	rneqd 4986	. 2 |- ( ( V e. W /\ E e. X ) -> ran (iEdg ` G ) = ran E )
43	40, 42	eqtrd 2265	1 |- ( ( V e. W /\ E e. X ) -> (Edg ` G ) = ran E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   =/= wne 2412   _Vcvv 2813   \ cdif 3208   C_ wss 3211   (/)c0 3508   {csn 3689   {cpr 3690   <.cop 3692   dom cdm 4749   ran crn 4750   Fun wfun 5346   Fn wfn 5347   ` cfv 5352   NNcn 9237   ndxcnx 13209   Basecbs 13212  .efcedgf 15999  iEdgciedg 16008  Edgcedg 16052

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-2nd 6335  df-1o 6647  df-2o 6648  df-en 6976  df-dom 6977  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-z 9578  df-dec 9710  df-uz 9854  df-fz 10343  df-struct 13214  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-iedg 16010  df-edg 16053

This theorem is referenced by: (None)