Theorem edgstruct 16510

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 16504	. . 3 |- Edg = ( g e. _V |-> ran (iEdg ` g ) )
2		fveq2 5698	. . . 4 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
3	2	rneqd 5013	. . 3 |- ( g = G -> ran (iEdg ` g ) = ran (iEdg ` G ) )
4		edgstruct.s	. . . 4 |- G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }
5		basendxnn 13466	. . . . . 6 |- ( Base ` ndx ) e. NN
6		simpl 109	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> V e. W )
7		opexg 4370	. . . . . 6 |- ( ( ( Base ` ndx ) e. NN /\ V e. W ) -> <. ( Base ` ndx ) , V >. e. _V )
8	5, 6, 7	sylancr 418	. . . . 5 |- ( ( V e. W /\ E e. X ) -> <. ( Base ` ndx ) , V >. e. _V )
9		edgfndxnn 16454	. . . . . 6 |- (.ef ` ndx ) e. NN
10		simpr 110	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> E e. X )
11		opexg 4370	. . . . . 6 |- ( ( (.ef ` ndx ) e. NN /\ E e. X ) -> <. (.ef ` ndx ) , E >. e. _V )
12	9, 10, 11	sylancr 418	. . . . 5 |- ( ( V e. W /\ E e. X ) -> <. (.ef ` ndx ) , E >. e. _V )
13		prexg 4351	. . . . 5 |- ( ( <. ( Base ` ndx ) , V >. e. _V /\ <. (.ef ` ndx ) , E >. e. _V ) -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } e. _V )
14	8, 12, 13	syl2anc 415	. . . 4 |- ( ( V e. W /\ E e. X ) -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } e. _V )
15	4, 14	eqeltrid 2325	. . 3 |- ( ( V e. W /\ E e. X ) -> G e. _V )
16	5	elexi 2834	. . . . . 6 |- ( Base ` ndx ) e. _V
17	9	elexi 2834	. . . . . 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 16457	. . . . . . . . . 10 |- ( Base ` ndx ) =/= (.ef ` ndx )
21	20	a1i 9	. . . . . . . . 9 |- ( ( V e. W /\ E e. X ) -> ( Base ` ndx ) =/= (.ef ` ndx ) )
22		fnprg 5439	. . . . . . . . 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 1289	. . . . . . . 8 |- ( ( V e. W /\ E e. X ) -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } Fn { ( Base ` ndx ) , (.ef ` ndx ) } )
24	4	fneq1i 5478	. . . . . . . 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 5481	. . . . . . 7 |- ( G Fn { ( Base ` ndx ) , (.ef ` ndx ) } -> Fun G )
27		fundif 5428	. . . . . . 7 |- ( Fun G -> Fun ( G \ { (/) } ) )
28	25, 26, 27	3syl 17	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> Fun ( G \ { (/) } ) )
29	25	fndmd 5485	. . . . . . 7 |- ( ( V e. W /\ E e. X ) -> dom G = { ( Base ` ndx ) , (.ef ` ndx ) } )
30		eqimss2 3303	. . . . . . 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 16478	. . . . 5 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) = (.ef ` G ) )
33		edgfid 16452	. . . . . . . 8 |- .ef = Slot (.ef ` ndx )
34	33, 9	ndxslid 13435	. . . . . . 7 |- (.ef = Slot (.ef ` ndx ) /\ (.ef ` ndx ) e. NN )
35	34	slotex 13437	. . . . . 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 2315	. . . 4 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) e. _V )
38		rnexg 5049	. . . 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 5802	. 2 |- ( ( V e. W /\ E e. X ) -> (Edg ` G ) = ran (iEdg ` G ) )
41	4	struct2griedg 16492	. . 3 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) = E )
42	41	rneqd 5013	. 2 |- ( ( V e. W /\ E e. X ) -> ran (iEdg ` G ) = ran E )
43	40, 42	eqtrd 2271	1 |- ( ( V e. W /\ E e. X ) -> (Edg ` G ) = ran E )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   \ cdif 3217   C_ wss 3220   (/)c0 3520   {csn 3709   {cpr 3710   <.cop 3712   dom cdm 4776   ran crn 4777   Fun wfun 5374   Fn wfn 5375   ` cfv 5380   NNcn 9313   ndxcnx 13407   Basecbs 13410  .efcedgf 16450  iEdgciedg 16459  Edgcedg 16503

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-2nd 6379  df-1o 6691  df-2o 6692  df-en 7027  df-dom 7028  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9656  df-dec 9789  df-uz 9938  df-fz 10429  df-struct 13412  df-ndx 13413  df-slot 13414  df-base 13416  df-edgf 16451  df-iedg 16461  df-edg 16504

This theorem is used by: (None)