Theorem fneqeql 5572

Description: Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Assertion

Ref	Expression
fneqeql	|- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> dom ( F i^i G ) = A ) )

Proof of Theorem fneqeql

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqfnfv 5562	. . 3 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
2		eqcom 2159	. . . 4 |- ( { x e. A | ( F ` x ) = ( G ` x ) } = A <-> A = { x e. A | ( F ` x ) = ( G ` x ) } )
3		rabid2 2633	. . . 4 |- ( A = { x e. A | ( F ` x ) = ( G ` x ) } <-> A. x e. A ( F ` x ) = ( G ` x ) )
4	2, 3	bitri 183	. . 3 |- ( { x e. A | ( F ` x ) = ( G ` x ) } = A <-> A. x e. A ( F ` x ) = ( G ` x ) )
5	1, 4	bitr4di 197	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> { x e. A | ( F ` x ) = ( G ` x ) } = A ) )
6		fndmin 5571	. . 3 |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) = { x e. A | ( F ` x ) = ( G ` x ) } )
7	6	eqeq1d 2166	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( dom ( F i^i G ) = A <-> { x e. A | ( F ` x ) = ( G ` x ) } = A ) )
8	5, 7	bitr4d 190	1 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> dom ( F i^i G ) = A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   A.wral 2435   {crab 2439   i^i cin 3101   dom cdm 4583   Fn wfn 5162   ` cfv 5167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-iota 5132  df-fun 5169  df-fn 5170  df-fv 5175

This theorem is referenced by:  fneqeql2  5573  fnreseql  5574