Theorem edgstruct 15879

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 15874	. . 3 |- Edg = ( g e. _V |-> ran (iEdg ` g ) )
2		fveq2 5629	. . . 4 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
3	2	rneqd 4953	. . 3 |- ( g = G -> ran (iEdg ` g ) = ran (iEdg ` G ) )
4		edgstruct.s	. . . 4 |- G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }
5		basendxnn 13103	. . . . . 6 |- ( Base ` ndx ) e. NN
6		simpl 109	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> V e. W )
7		opexg 4314	. . . . . 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 15824	. . . . . 6 |- (.ef ` ndx ) e. NN
10		simpr 110	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> E e. X )
11		opexg 4314	. . . . . 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 4295	. . . . 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 2316	. . 3 |- ( ( V e. W /\ E e. X ) -> G e. _V )
16	5	elexi 2812	. . . . . 6 |- ( Base ` ndx ) e. _V
17	9	elexi 2812	. . . . . 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 15827	. . . . . . . . . 10 |- ( Base ` ndx ) =/= (.ef ` ndx )
21	20	a1i 9	. . . . . . . . 9 |- ( ( V e. W /\ E e. X ) -> ( Base ` ndx ) =/= (.ef ` ndx ) )
22		fnprg 5376	. . . . . . . . 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 1282	. . . . . . . 8 |- ( ( V e. W /\ E e. X ) -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } Fn { ( Base ` ndx ) , (.ef ` ndx ) } )
24	4	fneq1i 5415	. . . . . . . 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 5418	. . . . . . 7 |- ( G Fn { ( Base ` ndx ) , (.ef ` ndx ) } -> Fun G )
27		fundif 5365	. . . . . . 7 |- ( Fun G -> Fun ( G \ { (/) } ) )
28	25, 26, 27	3syl 17	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> Fun ( G \ { (/) } ) )
29	25	fndmd 5422	. . . . . . 7 |- ( ( V e. W /\ E e. X ) -> dom G = { ( Base ` ndx ) , (.ef ` ndx ) } )
30		eqimss2 3279	. . . . . . 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 15848	. . . . 5 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) = (.ef ` G ) )
33		edgfid 15822	. . . . . . . 8 |- .ef = Slot (.ef ` ndx )
34	33, 9	ndxslid 13072	. . . . . . 7 |- (.ef = Slot (.ef ` ndx ) /\ (.ef ` ndx ) e. NN )
35	34	slotex 13074	. . . . . 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 2306	. . . 4 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) e. _V )
38		rnexg 4989	. . . 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 5730	. 2 |- ( ( V e. W /\ E e. X ) -> (Edg ` G ) = ran (iEdg ` G ) )
41	4	struct2griedg 15862	. . 3 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) = E )
42	41	rneqd 4953	. 2 |- ( ( V e. W /\ E e. X ) -> ran (iEdg ` G ) = ran E )
43	40, 42	eqtrd 2262	1 |- ( ( V e. W /\ E e. X ) -> (Edg ` G ) = ran E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   =/= wne 2400   _Vcvv 2799   \ cdif 3194   C_ wss 3197   (/)c0 3491   {csn 3666   {cpr 3667   <.cop 3669   dom cdm 4719   ran crn 4720   Fun wfun 5312   Fn wfn 5313   ` cfv 5318   NNcn 9121   ndxcnx 13044   Basecbs 13047  .efcedgf 15820  iEdgciedg 15829  Edgcedg 15873

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-2nd 6293  df-1o 6568  df-2o 6569  df-en 6896  df-dom 6897  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-z 9458  df-dec 9590  df-uz 9734  df-fz 10217  df-struct 13049  df-ndx 13050  df-slot 13051  df-base 13053  df-edgf 15821  df-iedg 15831  df-edg 15874

This theorem is referenced by: (None)