Theorem snstriedgval 39138

Description: The set of indexed edges of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See iedgvalsnop 39142 for the (degenerated) case where V = ( Base ` ndx ). (Contributed by AV, 24-Sep-2020.)

Hypotheses

Ref	Expression
snstrvtxval.v	|- V e. X
snstrvtxval.g	|- G = { <. ( Base ` ndx ) , V >. }

Assertion

Ref	Expression
snstriedgval	|- ( V =/= ( Base ` ndx ) -> (iEdg ` G ) = (/) )

Proof of Theorem snstriedgval

Step	Hyp	Ref	Expression
1		snstrvtxval.g	. . . 4 |- G = { <. ( Base ` ndx ) , V >. }
2		snex 4641	. . . 4 |- { <. ( Base ` ndx ) , V >. } e. _V
3	1, 2	eqeltri 2525	. . 3 |- G e. _V
4		iedgval 39106	. . 3 |- ( G e. _V -> (iEdg ` G ) = if ( G e. ( _V X. _V ) , ( 2nd ` G ) , (.ef ` G ) ) )
5	3, 4	mp1i 13	. 2 |- ( V =/= ( Base ` ndx ) -> (iEdg ` G ) = if ( G e. ( _V X. _V ) , ( 2nd ` G ) , (.ef ` G ) ) )
6		necom 2677	. . . 4 |- ( V =/= ( Base ` ndx ) <-> ( Base ` ndx ) =/= V )
7		fvex 5875	. . . . 5 |- ( Base ` ndx ) e. _V
8		snstrvtxval.v	. . . . 5 |- V e. X
9	7, 8, 1	funsndifnop 39022	. . . 4 |- ( ( Base ` ndx ) =/= V -> -. G e. ( _V X. _V ) )
10	6, 9	sylbi 199	. . 3 |- ( V =/= ( Base ` ndx ) -> -. G e. ( _V X. _V ) )
11	10	iffalsed 3892	. 2 |- ( V =/= ( Base ` ndx ) -> if ( G e. ( _V X. _V ) , ( 2nd ` G ) , (.ef ` G ) ) = (.ef ` G ) )
12	2	a1i 11	. . . . 5 |- ( G = { <. ( Base ` ndx ) , V >. } -> { <. ( Base ` ndx ) , V >. } e. _V )
13	1, 12	syl5eqel 2533	. . . 4 |- ( G = { <. ( Base ` ndx ) , V >. } -> G e. _V )
14		edgfndxid 39098	. . . 4 |- ( G e. _V -> (.ef ` G ) = ( G ` (.ef ` ndx ) ) )
15	1, 13, 14	mp2b 10	. . 3 |- (.ef ` G ) = ( G ` (.ef ` ndx ) )
16		slotsbaseefdif 39100	. . . . . . . 8 |- ( Base ` ndx ) =/= (.ef ` ndx )
17	16	nesymi 2681	. . . . . . 7 |- -. (.ef ` ndx ) = ( Base ` ndx )
18	17	a1i 11	. . . . . 6 |- ( V =/= ( Base ` ndx ) -> -. (.ef ` ndx ) = ( Base ` ndx ) )
19		fvex 5875	. . . . . . 7 |- (.ef ` ndx ) e. _V
20	19	elsnc 3992	. . . . . 6 |- ( (.ef ` ndx ) e. { ( Base ` ndx ) } <-> (.ef ` ndx ) = ( Base ` ndx ) )
21	18, 20	sylnibr 307	. . . . 5 |- ( V =/= ( Base ` ndx ) -> -. (.ef ` ndx ) e. { ( Base ` ndx ) } )
22	1	dmeqi 5036	. . . . . 6 |- dom G = dom { <. ( Base ` ndx ) , V >. }
23		dmsnopg 5307	. . . . . . 7 |- ( V e. X -> dom { <. ( Base ` ndx ) , V >. } = { ( Base ` ndx ) } )
24	8, 23	mp1i 13	. . . . . 6 |- ( V =/= ( Base ` ndx ) -> dom { <. ( Base ` ndx ) , V >. } = { ( Base ` ndx ) } )
25	22, 24	syl5eq 2497	. . . . 5 |- ( V =/= ( Base ` ndx ) -> dom G = { ( Base ` ndx ) } )
26	21, 25	neleqtrrd 2551	. . . 4 |- ( V =/= ( Base ` ndx ) -> -. (.ef ` ndx ) e. dom G )
27		ndmfv 5889	. . . 4 |- ( -. (.ef ` ndx ) e. dom G -> ( G ` (.ef ` ndx ) ) = (/) )
28	26, 27	syl 17	. . 3 |- ( V =/= ( Base ` ndx ) -> ( G ` (.ef ` ndx ) ) = (/) )
29	15, 28	syl5eq 2497	. 2 |- ( V =/= ( Base ` ndx ) -> (.ef ` G ) = (/) )
30	5, 11, 29	3eqtrd 2489	1 |- ( V =/= ( Base ` ndx ) -> (iEdg ` G ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1444   e. wcel 1887   =/= wne 2622   _Vcvv 3045   (/)c0 3731   ifcif 3881   {csn 3968   <.cop 3974   X. cxp 4832   dom cdm 4834   ` cfv 5582   2ndc2nd 6792   ndxcnx 15118   Basecbs 15121  .efcedgf 39095  iEdgciedg 39102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-ndx 15124  df-slot 15125  df-base 15126  df-edgf 39096  df-iedg 39104

This theorem is referenced by: (None)