Theorem structiedg0val 16161

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 13352	. . . . . . 7 |- ( Base ` ndx ) e. NN
3		simpl 109	. . . . . . 7 |- ( ( V e. X /\ E e. Y ) -> V e. X )
4		opexg 4349	. . . . . . 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 4349	. . . . . . 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 4330	. . . . . 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 2321	. . . 4 |- ( ( V e. X /\ E e. Y ) -> G e. _V )
13		structvtxvallem.b	. . . . . 6 |- ( Base ` ndx ) < S
14	1, 13, 6	2strstrndx 13415	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> G Struct <. ( Base ` ndx ) , S >. )
15		structn0fun 13309	. . . . 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 16159	. . . 4 |- ( ( V e. X /\ E e. Y ) -> 2o ~<_ dom G )
18		funiedgdm2domval 16151	. . . 4 |- ( ( G e. _V /\ Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> (iEdg ` G ) = (.ef ` G ) )
19	12, 16, 17, 18	syl3anc 1274	. . 3 |- ( ( V e. X /\ E e. Y ) -> (iEdg ` G ) = (.ef ` G ) )
20	19	3adant3 1044	. 2 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> (iEdg ` G ) = (.ef ` G ) )
21		edgfndxid 16130	. . . 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 1044	. 2 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> (.ef ` G ) = ( G ` (.ef ` ndx ) ) )
24		edgfndxnn 16129	. . . 4 |- (.ef ` ndx ) e. NN
25	24	elexi 2828	. . 3 |- (.ef ` ndx ) e. _V
26		basendxnedgfndx 16132	. . . . . . . 8 |- ( Base ` ndx ) =/= (.ef ` ndx )
27	26	nesymi 2460	. . . . . . 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 2436	. . . . . . . 8 |- ( S =/= (.ef ` ndx ) -> -. S = (.ef ` ndx ) )
30	29	neqcomd 2239	. . . . . . 7 |- ( S =/= (.ef ` ndx ) -> -. (.ef ` ndx ) = S )
31	30	3ad2ant3 1047	. . . . . 6 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> -. (.ef ` ndx ) = S )
32		ioran 760	. . . . . 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 3715	. . . . 5 |- ( (.ef ` ndx ) e. { ( Base ` ndx ) , S } <-> ( (.ef ` ndx ) = ( Base ` ndx ) \/ (.ef ` ndx ) = S ) )
35	33, 34	sylnibr 684	. . . 4 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> -. (.ef ` ndx ) e. { ( Base ` ndx ) , S } )
36	1	dmeqi 4962	. . . . 5 |- dom G = dom { <. ( Base ` ndx ) , V >. , <. S , E >. }
37		dmpropg 5240	. . . . . 6 |- ( ( V e. X /\ E e. Y ) -> dom { <. ( Base ` ndx ) , V >. , <. S , E >. } = { ( Base ` ndx ) , S } )
38	37	3adant3 1044	. . . . 5 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> dom { <. ( Base ` ndx ) , V >. , <. S , E >. } = { ( Base ` ndx ) , S } )
39	36, 38	eqtrid 2279	. . . 4 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> dom G = { ( Base ` ndx ) , S } )
40	35, 39	neleqtrrd 2333	. . 3 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> -. (.ef ` ndx ) e. dom G )
41		ndmfvg 5706	. . 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 2271	1 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> (iEdg ` G ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   _Vcvv 2815   \ cdif 3211   (/)c0 3512   {csn 3694   {cpr 3695   <.cop 3697   class class class wbr 4114   dom cdm 4754   Fun wfun 5351   ` cfv 5357   2oc2o 6654   ~<_ cdom 6987   < clt 8324   NNcn 9254   Struct cstr 13292   ndxcnx 13293   Basecbs 13296  .efcedgf 16125  iEdgciedg 16134

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

This theorem is referenced by: (None)