Theorem fnelfp 6441

Description: Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
fnelfp	|- ( ( F Fn A /\ X e. A ) -> ( X e. dom ( F i^i _I ) <-> ( F ` X ) = X ) )

Proof of Theorem fnelfp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fninfp 6440	. . 3 |- ( F Fn A -> dom ( F i^i _I ) = { x e. A | ( F ` x ) = x } )
2	1	eleq2d 2687	. 2 |- ( F Fn A -> ( X e. dom ( F i^i _I ) <-> X e. { x e. A | ( F ` x ) = x } ) )
3		fveq2 6191	. . . 4 |- ( x = X -> ( F ` x ) = ( F ` X ) )
4		id 22	. . . 4 |- ( x = X -> x = X )
5	3, 4	eqeq12d 2637	. . 3 |- ( x = X -> ( ( F ` x ) = x <-> ( F ` X ) = X ) )
6	5	elrab3 3364	. 2 |- ( X e. A -> ( X e. { x e. A | ( F ` x ) = x } <-> ( F ` X ) = X ) )
7	2, 6	sylan9bb 736	1 |- ( ( F Fn A /\ X e. A ) -> ( X e. dom ( F i^i _I ) <-> ( F ` X ) = X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   i^i cin 3573   _I cid 5023   dom cdm 5114   Fn wfn 5883   ` cfv 5888

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-sep 4781  ax-nul 4789  ax-pr 4906

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-mo 2475  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-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  ismrcd1  37261  ismrcd2  37262  istopclsd  37263