Theorem structiedg0val 16486

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 13466	. . . . . . 7 |- ( Base ` ndx ) e. NN
3		simpl 109	. . . . . . 7 |- ( ( V e. X /\ E e. Y ) -> V e. X )
4		opexg 4370	. . . . . . 7 |- ( ( ( Base ` ndx ) e. NN /\ V e. X ) -> <. ( Base ` ndx ) , V >. e. _V )
5	2, 3, 4	sylancr 418	. . . . . 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 4370	. . . . . . 7 |- ( ( S e. NN /\ E e. Y ) -> <. S , E >. e. _V )
9	6, 7, 8	sylancr 418	. . . . . 6 |- ( ( V e. X /\ E e. Y ) -> <. S , E >. e. _V )
10		prexg 4351	. . . . . 6 |- ( ( <. ( Base ` ndx ) , V >. e. _V /\ <. S , E >. e. _V ) -> { <. ( Base ` ndx ) , V >. , <. S , E >. } e. _V )
11	5, 9, 10	syl2anc 415	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> { <. ( Base ` ndx ) , V >. , <. S , E >. } e. _V )
12	1, 11	eqeltrid 2325	. . . 4 |- ( ( V e. X /\ E e. Y ) -> G e. _V )
13		structvtxvallem.b	. . . . . 6 |- ( Base ` ndx ) < S
14	1, 13, 6	2strstrndx 13531	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> G Struct <. ( Base ` ndx ) , S >. )
15		structn0fun 13423	. . . . 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 16484	. . . 4 |- ( ( V e. X /\ E e. Y ) -> 2o ~<_ dom G )
18		funiedgdm2domval 16476	. . . 4 |- ( ( G e. _V /\ Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> (iEdg ` G ) = (.ef ` G ) )
19	12, 16, 17, 18	syl3anc 1278	. . 3 |- ( ( V e. X /\ E e. Y ) -> (iEdg ` G ) = (.ef ` G ) )
20	19	3adant3 1048	. 2 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> (iEdg ` G ) = (.ef ` G ) )
21		edgfndxid 16455	. . . 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 1048	. 2 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> (.ef ` G ) = ( G ` (.ef ` ndx ) ) )
24		edgfndxnn 16454	. . . 4 |- (.ef ` ndx ) e. NN
25	24	elexi 2834	. . 3 |- (.ef ` ndx ) e. _V
26		basendxnedgfndx 16457	. . . . . . . 8 |- ( Base ` ndx ) =/= (.ef ` ndx )
27	26	nesymi 2466	. . . . . . 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 2442	. . . . . . . 8 |- ( S =/= (.ef ` ndx ) -> -. S = (.ef ` ndx ) )
30	29	neqcomd 2243	. . . . . . 7 |- ( S =/= (.ef ` ndx ) -> -. (.ef ` ndx ) = S )
31	30	3ad2ant3 1051	. . . . . 6 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> -. (.ef ` ndx ) = S )
32		ioran 764	. . . . . 6 |- ( -. ( (.ef ` ndx ) = ( Base ` ndx ) \/ (.ef ` ndx ) = S ) <-> ( -. (.ef ` ndx ) = ( Base ` ndx ) /\ -. (.ef ` ndx ) = S ) )
33	28, 31, 32	sylanbrc 421	. . . . 5 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> -. ( (.ef ` ndx ) = ( Base ` ndx ) \/ (.ef ` ndx ) = S ) )
34	25	elpr 3730	. . . . 5 |- ( (.ef ` ndx ) e. { ( Base ` ndx ) , S } <-> ( (.ef ` ndx ) = ( Base ` ndx ) \/ (.ef ` ndx ) = S ) )
35	33, 34	sylnibr 688	. . . 4 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> -. (.ef ` ndx ) e. { ( Base ` ndx ) , S } )
36	1	dmeqi 4984	. . . . 5 |- dom G = dom { <. ( Base ` ndx ) , V >. , <. S , E >. }
37		dmpropg 5263	. . . . . 6 |- ( ( V e. X /\ E e. Y ) -> dom { <. ( Base ` ndx ) , V >. , <. S , E >. } = { ( Base ` ndx ) , S } )
38	37	3adant3 1048	. . . . 5 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> dom { <. ( Base ` ndx ) , V >. , <. S , E >. } = { ( Base ` ndx ) , S } )
39	36, 38	eqtrid 2283	. . . 4 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> dom G = { ( Base ` ndx ) , S } )
40	35, 39	neleqtrrd 2337	. . 3 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> -. (.ef ` ndx ) e. dom G )
41		ndmfvg 5729	. . 3 |- ( ( (.ef ` ndx ) e. _V /\ -. (.ef ` ndx ) e. dom G ) -> ( G ` (.ef ` ndx ) ) = (/) )
42	25, 40, 41	sylancr 418	. 2 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> ( G ` (.ef ` ndx ) ) = (/) )
43	20, 23, 42	3eqtrd 2275	1 |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> (iEdg ` G ) = (/) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   \ cdif 3217   (/)c0 3520   {csn 3709   {cpr 3710   <.cop 3712   class class class wbr 4131   dom cdm 4776   Fun wfun 5374   ` cfv 5380   2oc2o 6685   ~<_ cdom 7025   < clt 8366   NNcn 9313   Struct cstr 13406   ndxcnx 13407   Basecbs 13410  .efcedgf 16450  iEdgciedg 16459

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

This theorem is used by: (None)