Theorem structiedg0val 15835

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 13083	. . . . . . 7 |- ( Base ` ndx ) e. NN
3		simpl 109	. . . . . . 7 |- ( ( V e. X /\ E e. Y ) -> V e. X )
4		opexg 4313	. . . . . . 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 4313	. . . . . . 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 4294	. . . . . 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 13146	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> G Struct <. ( Base ` ndx ) , S >. )
15		structn0fun 13040	. . . . 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 15833	. . . 4 |- ( ( V e. X /\ E e. Y ) -> 2o ~<_ dom G )
18		funiedgdm2domval 15825	. . . 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 15804	. . . 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 15803	. . . 4 |- (.ef ` ndx ) e. NN
25	24	elexi 2812	. . 3 |- (.ef ` ndx ) e. _V
26		basendxnedgfndx 15806	. . . . . . . 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 3687	. . . . 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 4923	. . . . 5 |- dom G = dom { <. ( Base ` ndx ) , V >. , <. S , E >. }
37		dmpropg 5200	. . . . . 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 5657	. . 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 2799   \ cdif 3194   (/)c0 3491   {csn 3666   {cpr 3667   <.cop 3669   class class class wbr 4082   dom cdm 4718   Fun wfun 5311   ` cfv 5317   2oc2o 6554   ~<_ cdom 6884   < clt 8177   NNcn 9106   Struct cstr 13023   ndxcnx 13024   Basecbs 13027  .efcedgf 15799  iEdgciedg 15808

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 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112

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 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-2nd 6285  df-1o 6560  df-2o 6561  df-en 6886  df-dom 6887  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-9 9172  df-n0 9366  df-z 9443  df-dec 9575  df-uz 9719  df-fz 10201  df-struct 13029  df-ndx 13030  df-slot 13031  df-base 13033  df-edgf 15800  df-iedg 15810

This theorem is referenced by: (None)