Theorem ispconn 31205

Description: The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypothesis

Ref	Expression
ispconn.1	|- X = U. J

Assertion

Ref	Expression
ispconn	|- ( J e. PConn <-> ( J e. Top /\ A. x e. X A. y e. X E. f e. ( II Cn J ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) ) )

Distinct variable groups:   x, f, y, J   x, X, y

Allowed substitution hint:   X( f)

Proof of Theorem ispconn

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4444	. . . 4 |- ( j = J -> U. j = U. J )
2		ispconn.1	. . . 4 |- X = U. J
3	1, 2	syl6eqr 2674	. . 3 |- ( j = J -> U. j = X )
4		oveq2 6658	. . . . 5 |- ( j = J -> ( II Cn j ) = ( II Cn J ) )
5	4	rexeqdv 3145	. . . 4 |- ( j = J -> ( E. f e. ( II Cn j ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) <-> E. f e. ( II Cn J ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) ) )
6	3, 5	raleqbidv 3152	. . 3 |- ( j = J -> ( A. y e. U. j E. f e. ( II Cn j ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) <-> A. y e. X E. f e. ( II Cn J ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) ) )
7	3, 6	raleqbidv 3152	. 2 |- ( j = J -> ( A. x e. U. j A. y e. U. j E. f e. ( II Cn j ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) <-> A. x e. X A. y e. X E. f e. ( II Cn J ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) ) )
8		df-pconn 31203	. 2 |- PConn = { j e. Top | A. x e. U. j A. y e. U. j E. f e. ( II Cn j ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) }
9	7, 8	elrab2 3366	1 |- ( J e. PConn <-> ( J e. Top /\ A. x e. X A. y e. X E. f e. ( II Cn J ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   U.cuni 4436   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   Topctop 20698   Cn ccn 21028   IIcii 22678  PConncpconn 31201

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-pconn 31203

This theorem is referenced by:  pconncn  31206  pconntop  31207  cnpconn  31212  txpconn  31214  ptpconn  31215  indispconn  31216  connpconn  31217  cvxpconn  31224