Theorem edgstruct 16185

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 16179	. . 3 |- Edg = ( g e. _V |-> ran (iEdg ` g ) )
2		fveq2 5675	. . . 4 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
3	2	rneqd 4991	. . 3 |- ( g = G -> ran (iEdg ` g ) = ran (iEdg ` G ) )
4		edgstruct.s	. . . 4 |- G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }
5		basendxnn 13352	. . . . . 6 |- ( Base ` ndx ) e. NN
6		simpl 109	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> V e. W )
7		opexg 4349	. . . . . 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 16129	. . . . . 6 |- (.ef ` ndx ) e. NN
10		simpr 110	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> E e. X )
11		opexg 4349	. . . . . 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 4330	. . . . 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 2321	. . 3 |- ( ( V e. W /\ E e. X ) -> G e. _V )
16	5	elexi 2828	. . . . . 6 |- ( Base ` ndx ) e. _V
17	9	elexi 2828	. . . . . 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 16132	. . . . . . . . . 10 |- ( Base ` ndx ) =/= (.ef ` ndx )
21	20	a1i 9	. . . . . . . . 9 |- ( ( V e. W /\ E e. X ) -> ( Base ` ndx ) =/= (.ef ` ndx ) )
22		fnprg 5416	. . . . . . . . 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 5455	. . . . . . . 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 5458	. . . . . . 7 |- ( G Fn { ( Base ` ndx ) , (.ef ` ndx ) } -> Fun G )
27		fundif 5405	. . . . . . 7 |- ( Fun G -> Fun ( G \ { (/) } ) )
28	25, 26, 27	3syl 17	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> Fun ( G \ { (/) } ) )
29	25	fndmd 5462	. . . . . . 7 |- ( ( V e. W /\ E e. X ) -> dom G = { ( Base ` ndx ) , (.ef ` ndx ) } )
30		eqimss2 3297	. . . . . . 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 16153	. . . . 5 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) = (.ef ` G ) )
33		edgfid 16127	. . . . . . . 8 |- .ef = Slot (.ef ` ndx )
34	33, 9	ndxslid 13321	. . . . . . 7 |- (.ef = Slot (.ef ` ndx ) /\ (.ef ` ndx ) e. NN )
35	34	slotex 13323	. . . . . 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 2311	. . . 4 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) e. _V )
38		rnexg 5027	. . . 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 5776	. 2 |- ( ( V e. W /\ E e. X ) -> (Edg ` G ) = ran (iEdg ` G ) )
41	4	struct2griedg 16167	. . 3 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) = E )
42	41	rneqd 4991	. 2 |- ( ( V e. W /\ E e. X ) -> ran (iEdg ` G ) = ran E )
43	40, 42	eqtrd 2267	1 |- ( ( V e. W /\ E e. X ) -> (Edg ` G ) = ran E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   =/= wne 2414   _Vcvv 2815   \ cdif 3211   C_ wss 3214   (/)c0 3512   {csn 3694   {cpr 3695   <.cop 3697   dom cdm 4754   ran crn 4755   Fun wfun 5351   Fn wfn 5352   ` cfv 5357   NNcn 9254   ndxcnx 13293   Basecbs 13296  .efcedgf 16125  iEdgciedg 16134  Edgcedg 16178

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-2nd 6348  df-1o 6660  df-2o 6661  df-en 6989  df-dom 6990  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-fz 10362  df-struct 13298  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-iedg 16136  df-edg 16179

This theorem is referenced by: (None)