Theorem snstriedgval 39291

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 39295 for the (degenerated) case where V = ( Base ` ndx ). (Contributed by AV, 24-Sep-2020.)

Hypotheses

Ref	Expression
snstrvtxval.v	|- V e. _V
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 2545	. . 3 |- G e. _V
4		iedgval 39256	. . 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 2696	. . . 4 |- ( V =/= ( Base ` ndx ) <-> ( Base ` ndx ) =/= V )
7		fvex 5889	. . . . 5 |- ( Base ` ndx ) e. _V
8		snstrvtxval.v	. . . . 5 |- V e. _V
9	7, 8, 1	funsndifnop 39168	. . . 4 |- ( ( Base ` ndx ) =/= V -> -. G e. ( _V X. _V ) )
10	6, 9	sylbi 200	. . 3 |- ( V =/= ( Base ` ndx ) -> -. G e. ( _V X. _V ) )
11	10	iffalsed 3883	. 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 2553	. . . 4 |- ( G = { <. ( Base ` ndx ) , V >. } -> G e. _V )
14		edgfndxid 39248	. . . 4 |- ( G e. _V -> (.ef ` G ) = ( G ` (.ef ` ndx ) ) )
15	1, 13, 14	mp2b 10	. . 3 |- (.ef ` G ) = ( G ` (.ef ` ndx ) )
16		slotsbaseefdif 39250	. . . . . . . 8 |- ( Base ` ndx ) =/= (.ef ` ndx )
17	16	nesymi 2700	. . . . . . 7 |- -. (.ef ` ndx ) = ( Base ` ndx )
18	17	a1i 11	. . . . . 6 |- ( V =/= ( Base ` ndx ) -> -. (.ef ` ndx ) = ( Base ` ndx ) )
19		fvex 5889	. . . . . . 7 |- (.ef ` ndx ) e. _V
20	19	elsnc 3984	. . . . . 6 |- ( (.ef ` ndx ) e. { ( Base ` ndx ) } <-> (.ef ` ndx ) = ( Base ` ndx ) )
21	18, 20	sylnibr 312	. . . . 5 |- ( V =/= ( Base ` ndx ) -> -. (.ef ` ndx ) e. { ( Base ` ndx ) } )
22	1	dmeqi 5041	. . . . . 6 |- dom G = dom { <. ( Base ` ndx ) , V >. }
23		dmsnopg 5314	. . . . . . 7 |- ( V e. _V -> 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 2517	. . . . 5 |- ( V =/= ( Base ` ndx ) -> dom G = { ( Base ` ndx ) } )
26	21, 25	neleqtrrd 2571	. . . 4 |- ( V =/= ( Base ` ndx ) -> -. (.ef ` ndx ) e. dom G )
27		ndmfv 5903	. . . 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 2517	. 2 |- ( V =/= ( Base ` ndx ) -> (.ef ` G ) = (/) )
30	5, 11, 29	3eqtrd 2509	1 |- ( V =/= ( Base ` ndx ) -> (iEdg ` G ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1452   e. wcel 1904   =/= wne 2641   _Vcvv 3031   (/)c0 3722   ifcif 3872   {csn 3959   <.cop 3965   X. cxp 4837   dom cdm 4839   ` cfv 5589   2ndc2nd 6811   ndxcnx 15196   Basecbs 15199  .efcedgf 39245  iEdgciedg 39252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-ndx 15202  df-slot 15203  df-base 15204  df-edgf 39246  df-iedg 39254

This theorem is referenced by: (None)