Theorem pconncn 31206

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
pconncn	|- ( ( J e. PConn /\ A e. X /\ B e. X ) -> E. f e. ( II Cn J ) ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) )

Distinct variable groups:   A, f   B, f   f, J

Allowed substitution hint:   X( f)

Proof of Theorem pconncn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ispconn.1	. . . . 5 |- X = U. J
2	1	ispconn 31205	. . . 4 |- ( 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 ) ) )
3	2	simprbi 480	. . 3 |- ( J e. PConn -> A. x e. X A. y e. X E. f e. ( II Cn J ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) )
4		eqeq2 2633	. . . . . 6 |- ( x = A -> ( ( f ` 0 ) = x <-> ( f ` 0 ) = A ) )
5	4	anbi1d 741	. . . . 5 |- ( x = A -> ( ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) <-> ( ( f ` 0 ) = A /\ ( f ` 1 ) = y ) ) )
6	5	rexbidv 3052	. . . 4 |- ( x = A -> ( E. f e. ( II Cn J ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) <-> E. f e. ( II Cn J ) ( ( f ` 0 ) = A /\ ( f ` 1 ) = y ) ) )
7		eqeq2 2633	. . . . . 6 |- ( y = B -> ( ( f ` 1 ) = y <-> ( f ` 1 ) = B ) )
8	7	anbi2d 740	. . . . 5 |- ( y = B -> ( ( ( f ` 0 ) = A /\ ( f ` 1 ) = y ) <-> ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) )
9	8	rexbidv 3052	. . . 4 |- ( y = B -> ( E. f e. ( II Cn J ) ( ( f ` 0 ) = A /\ ( f ` 1 ) = y ) <-> E. f e. ( II Cn J ) ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) )
10	6, 9	rspc2v 3322	. . 3 |- ( ( A e. X /\ B e. X ) -> ( A. x e. X A. y e. X E. f e. ( II Cn J ) ( ( f ` 0 ) = x /\ ( f ` 1 ) = y ) -> E. f e. ( II Cn J ) ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) )
11	3, 10	syl5com 31	. 2 |- ( J e. PConn -> ( ( A e. X /\ B e. X ) -> E. f e. ( II Cn J ) ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) )
12	11	3impib 1262	1 |- ( ( J e. PConn /\ A e. X /\ B e. X ) -> E. f e. ( II Cn J ) ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = 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:  cnpconn  31212  pconnconn  31213  txpconn  31214  ptpconn  31215  connpconn  31217  pconnpi1  31219  cvmlift3lem2  31302  cvmlift3lem7  31307