Theorem structiedg0val 15881

Description: The set of indexed edges of an extensible structure with a base set and another slot not being the slot for edge functions is empty. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)

Hypotheses

Ref	Expression
structvtxvallem.s	|- S e. NN
structvtxvallem.b	|- ( Base ` ndx ) < S
structvtxvallem.g	|- G = { <. ( Base ` ndx ) , V >. , <. S , E >. }

Assertion

Ref	Expression
structiedg0val	|- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> (iEdg ` G ) = (/) )

Proof of Theorem structiedg0val

Step	Hyp	Ref	Expression
1		structvtxvallem.g	. . . . 5 |- G = { <. ( Base ` ndx ) , V >. , <. S , E >. }
2		basendxnn 13128	. . . . . . 7 |- ( Base ` ndx ) e. NN
3		simpl 109	. . . . . . 7 |- ( ( V e. X /\ E e. Y ) -> V e. X )
4		opexg 4318	. . . . . . 7 |- ( ( ( Base ` ndx ) e. NN /\ V e. X ) -> <. ( Base ` ndx ) , V >. e. _V )
5	2, 3, 4	sylancr 414	. . . . . 6 |- ( ( V e. X /\ E e. Y ) -> <. ( Base ` ndx ) , V >. e. _V )
6		structvtxvallem.s	. . . . . . 7 |- S e. NN
7		simpr 110	. . . . . . 7 |- ( ( V e. X /\ E e. Y ) -> E e. Y )
8		opexg 4318	. . . . . . 7 |- ( ( S e. NN /\ E e. Y ) -> <. S , E >. e. _V )
9	6, 7, 8	sylancr 414	. . . . . 6 |- ( ( V e. X /\ E e. Y ) -> <. S , E >. e. _V )
10		prexg 4299	. . . . . 6 |- ( ( <. ( Base ` ndx ) , V >. e. _V /\ <. S , E >. e. _V ) -> { <. ( Base ` ndx ) , V >. , <. S , E >. } e. _V )
11	5, 9, 10	syl2anc 411	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> { <. ( Base ` ndx ) , V >. , <. S , E >. } e. _V )
12	1, 11	eqeltrid 2316	. . . 4 |- ( ( V e. X /\ E e. Y ) -> G e. _V )
13		structvtxvallem.b	. . . . . 6 |- ( Base ` ndx ) < S
14	1, 13, 6	2strstrndx 13191	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> G Struct <. ( Base ` ndx ) , S >. )
15		structn0fun 13085	. . . . 5 |- ( G Struct <. ( Base ` ndx ) , S >. -> Fun ( G \ { (/) } ) )
16	14, 15	syl 14	. . . 4 |- ( ( V e. X /\ E e. Y ) -> Fun ( G \ { (/) } ) )
17	6, 13, 1	struct2slots2dom 15879	. . . 4 |- ( ( V e. X /\ E e. Y ) -> 2o ~<_ dom G )
18		funiedgdm2domval 15871	. . . 4 |- ( ( G e. _V /\ Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> (iEdg ` G ) = (.ef ` G ) )
19	12, 16, 17, 18	syl3anc 1271	. . 3 |- ( ( V e. X /\ E e. Y ) -> (iEdg ` G ) = (.ef ` G ) )
20	19	3adant3 1041	. 2 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> (iEdg ` G ) = (.ef ` G ) )
21		edgfndxid 15850	. . . 4 |- ( G e. _V -> (.ef ` G ) = ( G ` (.ef ` ndx ) ) )
22	12, 21	syl 14	. . 3 |- ( ( V e. X /\ E e. Y ) -> (.ef ` G ) = ( G ` (.ef ` ndx ) ) )
23	22	3adant3 1041	. 2 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> (.ef ` G ) = ( G ` (.ef ` ndx ) ) )
24		edgfndxnn 15849	. . . 4 |- (.ef ` ndx ) e. NN
25	24	elexi 2813	. . 3 |- (.ef ` ndx ) e. _V
26		basendxnedgfndx 15852	. . . . . . . 8 |- ( Base ` ndx ) =/= (.ef ` ndx )
27	26	nesymi 2446	. . . . . . 7 |- -. (.ef ` ndx ) = ( Base ` ndx )
28	27	a1i 9	. . . . . 6 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> -. (.ef ` ndx ) = ( Base ` ndx ) )
29		neneq 2422	. . . . . . . 8 |- ( S =/= (.ef ` ndx ) -> -. S = (.ef ` ndx ) )
30	29	neqcomd 2234	. . . . . . 7 |- ( S =/= (.ef ` ndx ) -> -. (.ef ` ndx ) = S )
31	30	3ad2ant3 1044	. . . . . 6 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> -. (.ef ` ndx ) = S )
32		ioran 757	. . . . . 6 |- ( -. ( (.ef ` ndx ) = ( Base ` ndx ) \/ (.ef ` ndx ) = S ) <-> ( -. (.ef ` ndx ) = ( Base ` ndx ) /\ -. (.ef ` ndx ) = S ) )
33	28, 31, 32	sylanbrc 417	. . . . 5 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> -. ( (.ef ` ndx ) = ( Base ` ndx ) \/ (.ef ` ndx ) = S ) )
34	25	elpr 3688	. . . . 5 |- ( (.ef ` ndx ) e. { ( Base ` ndx ) , S } <-> ( (.ef ` ndx ) = ( Base ` ndx ) \/ (.ef ` ndx ) = S ) )
35	33, 34	sylnibr 681	. . . 4 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> -. (.ef ` ndx ) e. { ( Base ` ndx ) , S } )
36	1	dmeqi 4930	. . . . 5 |- dom G = dom { <. ( Base ` ndx ) , V >. , <. S , E >. }
37		dmpropg 5207	. . . . . 6 |- ( ( V e. X /\ E e. Y ) -> dom { <. ( Base ` ndx ) , V >. , <. S , E >. } = { ( Base ` ndx ) , S } )
38	37	3adant3 1041	. . . . 5 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> dom { <. ( Base ` ndx ) , V >. , <. S , E >. } = { ( Base ` ndx ) , S } )
39	36, 38	eqtrid 2274	. . . 4 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> dom G = { ( Base ` ndx ) , S } )
40	35, 39	neleqtrrd 2328	. . 3 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> -. (.ef ` ndx ) e. dom G )
41		ndmfvg 5666	. . 3 |- ( ( (.ef ` ndx ) e. _V /\ -. (.ef ` ndx ) e. dom G ) -> ( G ` (.ef ` ndx ) ) = (/) )
42	25, 40, 41	sylancr 414	. 2 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> ( G ` (.ef ` ndx ) ) = (/) )
43	20, 23, 42	3eqtrd 2266	1 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> (iEdg ` G ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   _Vcvv 2800   \ cdif 3195   (/)c0 3492   {csn 3667   {cpr 3668   <.cop 3670   class class class wbr 4086   dom cdm 4723   Fun wfun 5318   ` cfv 5324   2oc2o 6571   ~<_ cdom 6903   < clt 8204   NNcn 9133   Struct cstr 13068   ndxcnx 13069   Basecbs 13072  .efcedgf 15845  iEdgciedg 15854

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 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-2nd 6299  df-1o 6577  df-2o 6578  df-en 6905  df-dom 6906  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-z 9470  df-dec 9602  df-uz 9746  df-fz 10234  df-struct 13074  df-ndx 13075  df-slot 13076  df-base 13078  df-edgf 15846  df-iedg 15856

This theorem is referenced by: (None)