Theorem isconngr 27049

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
isconngr	|- ( G e. W -> ( G e. ConnGraph <-> A. k e. V A. n e. V 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 isconngr

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

Step	Hyp	Ref	Expression
1		df-conngr 27047	. . 3 |- ConnGraph = { g | [. (Vtx ` g ) / v ]. A. k e. v A. n e. v 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 E. f E. p f ( k (PathsOn ` g ) n ) p } )
3		fvex 6201	. . . . . 6 |- (Vtx ` g ) e. _V
4		raleq 3138	. . . . . . 7 |- ( v = (Vtx ` g ) -> ( A. n e. v E. f E. p f ( k (PathsOn ` g ) n ) p <-> A. n e. (Vtx ` g ) E. f E. p f ( k (PathsOn ` g ) n ) p ) )
5	4	raleqbi1dv 3146	. . . . . 6 |- ( v = (Vtx ` g ) -> ( A. k e. v A. n e. v E. f E. p f ( k (PathsOn ` g ) n ) p <-> A. k e. (Vtx ` g ) A. n e. (Vtx ` g ) E. f E. p f ( k (PathsOn ` g ) n ) p ) )
6	3, 5	sbcie 3470	. . . . 5 |- ( [. (Vtx ` g ) / v ]. A. k e. v A. n e. v E. f E. p f ( k (PathsOn ` g ) n ) p <-> A. k e. (Vtx ` g ) A. n e. (Vtx ` g ) E. f E. p f ( k (PathsOn ` g ) n ) p )
7	6	abbii 2739	. . . 4 |- { g | [. (Vtx ` g ) / v ]. A. k e. v A. n e. v E. f E. p f ( k (PathsOn ` g ) n ) p } = { g | A. k e. (Vtx ` g ) A. n e. (Vtx ` g ) E. f E. p f ( k (PathsOn ` g ) n ) p }
8	7	eleq2i 2693	. . 3 |- ( G e. { g | [. (Vtx ` g ) / v ]. A. k e. v A. n e. v E. f E. p f ( k (PathsOn ` g ) n ) p } <-> G e. { g | A. k e. (Vtx ` g ) A. n e. (Vtx ` g ) E. f E. p f ( k (PathsOn ` g ) n ) p } )
9		fveq2 6191	. . . . . 6 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
10		isconngr.v	. . . . . 6 |- V = (Vtx ` G )
11	9, 10	syl6eqr 2674	. . . . 5 |- ( h = G -> (Vtx ` h ) = V )
12		fveq2 6191	. . . . . . . . 9 |- ( h = G -> (PathsOn ` h ) = (PathsOn ` G ) )
13	12	oveqd 6667	. . . . . . . 8 |- ( h = G -> ( k (PathsOn ` h ) n ) = ( k (PathsOn ` G ) n ) )
14	13	breqd 4664	. . . . . . 7 |- ( h = G -> ( f ( k (PathsOn ` h ) n ) p <-> f ( k (PathsOn ` G ) n ) p ) )
15	14	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 ) )
16	11, 15	raleqbidv 3152	. . . . 5 |- ( h = G -> ( A. n e. (Vtx ` h ) E. f E. p f ( k (PathsOn ` h ) n ) p <-> A. n e. V E. f E. p f ( k (PathsOn ` G ) n ) p ) )
17	11, 16	raleqbidv 3152	. . . 4 |- ( h = G -> ( A. k e. (Vtx ` h ) A. n e. (Vtx ` h ) E. f E. p f ( k (PathsOn ` h ) n ) p <-> A. k e. V A. n e. V E. f E. p f ( k (PathsOn ` G ) n ) p ) )
18		fveq2 6191	. . . . . 6 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
19		fveq2 6191	. . . . . . . . . 10 |- ( g = h -> (PathsOn ` g ) = (PathsOn ` h ) )
20	19	oveqd 6667	. . . . . . . . 9 |- ( g = h -> ( k (PathsOn ` g ) n ) = ( k (PathsOn ` h ) n ) )
21	20	breqd 4664	. . . . . . . 8 |- ( g = h -> ( f ( k (PathsOn ` g ) n ) p <-> f ( k (PathsOn ` h ) n ) p ) )
22	21	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 ) )
23	18, 22	raleqbidv 3152	. . . . . 6 |- ( g = h -> ( A. n e. (Vtx ` g ) E. f E. p f ( k (PathsOn ` g ) n ) p <-> A. n e. (Vtx ` h ) E. f E. p f ( k (PathsOn ` h ) n ) p ) )
24	18, 23	raleqbidv 3152	. . . . 5 |- ( g = h -> ( A. k e. (Vtx ` g ) A. n e. (Vtx ` g ) E. f E. p f ( k (PathsOn ` g ) n ) p <-> A. k e. (Vtx ` h ) A. n e. (Vtx ` h ) E. f E. p f ( k (PathsOn ` h ) n ) p ) )
25	24	cbvabv 2747	. . . 4 |- { g | A. k e. (Vtx ` g ) A. n e. (Vtx ` g ) E. f E. p f ( k (PathsOn ` g ) n ) p } = { h | A. k e. (Vtx ` h ) A. n e. (Vtx ` h ) E. f E. p f ( k (PathsOn ` h ) n ) p }
26	17, 25	elab2g 3353	. . 3 |- ( G e. W -> ( G e. { g | A. k e. (Vtx ` g ) A. n e. (Vtx ` g ) E. f E. p f ( k (PathsOn ` g ) n ) p } <-> A. k e. V A. n e. V E. f E. p f ( k (PathsOn ` G ) n ) p ) )
27	8, 26	syl5bb 272	. 2 |- ( G e. W -> ( G e. { g | [. (Vtx ` g ) / v ]. A. k e. v A. n e. v E. f E. p f ( k (PathsOn ` g ) n ) p } <-> A. k e. V A. n e. V E. f E. p f ( k (PathsOn ` G ) n ) p ) )
28	2, 27	syl5bb 272	1 |- ( G e. W -> ( G e. ConnGraph <-> A. k e. V A. n e. V 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   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-conngr 27047

This theorem is referenced by:  0conngr  27052  0vconngr  27053  1conngr  27054  conngrv2edg  27055