Theorem edgstruct 15745

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 15740	. . 3 |- Edg = ( g e. _V |-> ran (iEdg ` g ) )
2		fveq2 5594	. . . 4 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
3	2	rneqd 4921	. . 3 |- ( g = G -> ran (iEdg ` g ) = ran (iEdg ` G ) )
4		edgstruct.s	. . . 4 |- G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }
5		basendxnn 12973	. . . . . 6 |- ( Base ` ndx ) e. NN
6		simpl 109	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> V e. W )
7		opexg 4285	. . . . . 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 15692	. . . . . 6 |- (.ef ` ndx ) e. NN
10		simpr 110	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> E e. X )
11		opexg 4285	. . . . . 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 4266	. . . . 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 2293	. . 3 |- ( ( V e. W /\ E e. X ) -> G e. _V )
16	5	elexi 2786	. . . . . 6 |- ( Base ` ndx ) e. _V
17	9	elexi 2786	. . . . . 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 15695	. . . . . . . . . 10 |- ( Base ` ndx ) =/= (.ef ` ndx )
21	20	a1i 9	. . . . . . . . 9 |- ( ( V e. W /\ E e. X ) -> ( Base ` ndx ) =/= (.ef ` ndx ) )
22		fnprg 5343	. . . . . . . . 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 1261	. . . . . . . 8 |- ( ( V e. W /\ E e. X ) -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } Fn { ( Base ` ndx ) , (.ef ` ndx ) } )
24	4	fneq1i 5382	. . . . . . . 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 5385	. . . . . . 7 |- ( G Fn { ( Base ` ndx ) , (.ef ` ndx ) } -> Fun G )
27		fundif 5332	. . . . . . 7 |- ( Fun G -> Fun ( G \ { (/) } ) )
28	25, 26, 27	3syl 17	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> Fun ( G \ { (/) } ) )
29	25	fndmd 5389	. . . . . . 7 |- ( ( V e. W /\ E e. X ) -> dom G = { ( Base ` ndx ) , (.ef ` ndx ) } )
30		eqimss2 3252	. . . . . . 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 15716	. . . . 5 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) = (.ef ` G ) )
33		edgfid 15690	. . . . . . . 8 |- .ef = Slot (.ef ` ndx )
34	33, 9	ndxslid 12942	. . . . . . 7 |- (.ef = Slot (.ef ` ndx ) /\ (.ef ` ndx ) e. NN )
35	34	slotex 12944	. . . . . 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 2283	. . . 4 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) e. _V )
38		rnexg 4957	. . . 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 5691	. 2 |- ( ( V e. W /\ E e. X ) -> (Edg ` G ) = ran (iEdg ` G ) )
41	4	struct2griedg 15730	. . 3 |- ( ( V e. W /\ E e. X ) -> (iEdg ` G ) = E )
42	41	rneqd 4921	. 2 |- ( ( V e. W /\ E e. X ) -> ran (iEdg ` G ) = ran E )
43	40, 42	eqtrd 2239	1 |- ( ( V e. W /\ E e. X ) -> (Edg ` G ) = ran E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   =/= wne 2377   _Vcvv 2773   \ cdif 3167   C_ wss 3170   (/)c0 3464   {csn 3638   {cpr 3639   <.cop 3641   dom cdm 4688   ran crn 4689   Fun wfun 5279   Fn wfn 5280   ` cfv 5285   NNcn 9066   ndxcnx 12914   Basecbs 12917  .efcedgf 15688  iEdgciedg 15697  Edgcedg 15739

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-2nd 6245  df-1o 6520  df-2o 6521  df-en 6846  df-dom 6847  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-inn 9067  df-2 9125  df-3 9126  df-4 9127  df-5 9128  df-6 9129  df-7 9130  df-8 9131  df-9 9132  df-n0 9326  df-z 9403  df-dec 9535  df-uz 9679  df-fz 10161  df-struct 12919  df-ndx 12920  df-slot 12921  df-base 12923  df-edgf 15689  df-iedg 15699  df-edg 15740

This theorem is referenced by: (None)