Theorem isconngr1 27050

Description: The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)

Hypothesis

Ref	Expression
isconngr.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
isconngr1	|- ( G e. W -> ( G e. ConnGraph <-> A. k e. V A. n e. ( V \ { k } ) E. f E. p f ( k (PathsOn ` G ) n ) p ) )

Distinct variable groups:   f, k, n, p, G   k, V, n

Allowed substitution hints:   V( f, p)   W( f, k, n, p)

Proof of Theorem isconngr1

Dummy variables g h v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfconngr1 27048	. . 3 |- ConnGraph = { g | [. (Vtx ` g ) / v ]. A. k e. v A. n e. ( v \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p }
2	1	eleq2i 2693	. 2 |- ( G e. ConnGraph <-> G e. { g | [. (Vtx ` g ) / v ]. A. k e. v A. n e. ( v \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p } )
3		fvex 6201	. . . . . 6 |- (Vtx ` g ) e. _V
4		id 22	. . . . . . 7 |- ( v = (Vtx ` g ) -> v = (Vtx ` g ) )
5		difeq1 3721	. . . . . . . 8 |- ( v = (Vtx ` g ) -> ( v \ { k } ) = ( (Vtx ` g ) \ { k } ) )
6	5	raleqdv 3144	. . . . . . 7 |- ( v = (Vtx ` g ) -> ( A. n e. ( v \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p <-> A. n e. ( (Vtx ` g ) \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p ) )
7	4, 6	raleqbidv 3152	. . . . . 6 |- ( v = (Vtx ` g ) -> ( A. k e. v A. n e. ( v \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p <-> A. k e. (Vtx ` g ) A. n e. ( (Vtx ` g ) \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p ) )
8	3, 7	sbcie 3470	. . . . 5 |- ( [. (Vtx ` g ) / v ]. A. k e. v A. n e. ( v \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p <-> A. k e. (Vtx ` g ) A. n e. ( (Vtx ` g ) \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p )
9	8	abbii 2739	. . . 4 |- { g | [. (Vtx ` g ) / v ]. A. k e. v A. n e. ( v \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p } = { g | A. k e. (Vtx ` g ) A. n e. ( (Vtx ` g ) \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p }
10	9	eleq2i 2693	. . 3 |- ( G e. { g | [. (Vtx ` g ) / v ]. A. k e. v A. n e. ( v \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p } <-> G e. { g | A. k e. (Vtx ` g ) A. n e. ( (Vtx ` g ) \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p } )
11		fveq2 6191	. . . . . 6 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
12		isconngr.v	. . . . . 6 |- V = (Vtx ` G )
13	11, 12	syl6eqr 2674	. . . . 5 |- ( h = G -> (Vtx ` h ) = V )
14	13	difeq1d 3727	. . . . . 6 |- ( h = G -> ( (Vtx ` h ) \ { k } ) = ( V \ { k } ) )
15		fveq2 6191	. . . . . . . . 9 |- ( h = G -> (PathsOn ` h ) = (PathsOn ` G ) )
16	15	oveqd 6667	. . . . . . . 8 |- ( h = G -> ( k (PathsOn ` h ) n ) = ( k (PathsOn ` G ) n ) )
17	16	breqd 4664	. . . . . . 7 |- ( h = G -> ( f ( k (PathsOn ` h ) n ) p <-> f ( k (PathsOn ` G ) n ) p ) )
18	17	2exbidv 1852	. . . . . 6 |- ( h = G -> ( E. f E. p f ( k (PathsOn ` h ) n ) p <-> E. f E. p f ( k (PathsOn ` G ) n ) p ) )
19	14, 18	raleqbidv 3152	. . . . 5 |- ( h = G -> ( A. n e. ( (Vtx ` h ) \ { k } ) E. f E. p f ( k (PathsOn ` h ) n ) p <-> A. n e. ( V \ { k } ) E. f E. p f ( k (PathsOn ` G ) n ) p ) )
20	13, 19	raleqbidv 3152	. . . 4 |- ( h = G -> ( A. k e. (Vtx ` h ) A. n e. ( (Vtx ` h ) \ { k } ) E. f E. p f ( k (PathsOn ` h ) n ) p <-> A. k e. V A. n e. ( V \ { k } ) E. f E. p f ( k (PathsOn ` G ) n ) p ) )
21		fveq2 6191	. . . . . 6 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
22	21	difeq1d 3727	. . . . . . 7 |- ( g = h -> ( (Vtx ` g ) \ { k } ) = ( (Vtx ` h ) \ { k } ) )
23		fveq2 6191	. . . . . . . . . 10 |- ( g = h -> (PathsOn ` g ) = (PathsOn ` h ) )
24	23	oveqd 6667	. . . . . . . . 9 |- ( g = h -> ( k (PathsOn ` g ) n ) = ( k (PathsOn ` h ) n ) )
25	24	breqd 4664	. . . . . . . 8 |- ( g = h -> ( f ( k (PathsOn ` g ) n ) p <-> f ( k (PathsOn ` h ) n ) p ) )
26	25	2exbidv 1852	. . . . . . 7 |- ( g = h -> ( E. f E. p f ( k (PathsOn ` g ) n ) p <-> E. f E. p f ( k (PathsOn ` h ) n ) p ) )
27	22, 26	raleqbidv 3152	. . . . . 6 |- ( g = h -> ( A. n e. ( (Vtx ` g ) \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p <-> A. n e. ( (Vtx ` h ) \ { k } ) E. f E. p f ( k (PathsOn ` h ) n ) p ) )
28	21, 27	raleqbidv 3152	. . . . 5 |- ( g = h -> ( A. k e. (Vtx ` g ) A. n e. ( (Vtx ` g ) \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p <-> A. k e. (Vtx ` h ) A. n e. ( (Vtx ` h ) \ { k } ) E. f E. p f ( k (PathsOn ` h ) n ) p ) )
29	28	cbvabv 2747	. . . 4 |- { g | A. k e. (Vtx ` g ) A. n e. ( (Vtx ` g ) \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p } = { h | A. k e. (Vtx ` h ) A. n e. ( (Vtx ` h ) \ { k } ) E. f E. p f ( k (PathsOn ` h ) n ) p }
30	20, 29	elab2g 3353	. . 3 |- ( G e. W -> ( G e. { g | A. k e. (Vtx ` g ) A. n e. ( (Vtx ` g ) \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p } <-> A. k e. V A. n e. ( V \ { k } ) E. f E. p f ( k (PathsOn ` G ) n ) p ) )
31	10, 30	syl5bb 272	. 2 |- ( G e. W -> ( G e. { g | [. (Vtx ` g ) / v ]. A. k e. v A. n e. ( v \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p } <-> A. k e. V A. n e. ( V \ { k } ) E. f E. p f ( k (PathsOn ` G ) n ) p ) )
32	2, 31	syl5bb 272	1 |- ( G e. W -> ( G e. ConnGraph <-> A. k e. V A. n e. ( V \ { k } ) E. f E. p f ( k (PathsOn ` G ) n ) p ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   [.wsbc 3435   \ cdif 3571   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  PathsOncpthson 26610  ConnGraphcconngr 27046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-pthson 26614  df-conngr 27047

This theorem is referenced by:  cusconngr  27051  frgrconngr  27158