Theorem fneqeql 5786

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 5775	. . 3 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
2		eqcom 2234	. . . 4 |- ( { x e. A | ( F ` x ) = ( G ` x ) } = A <-> A = { x e. A | ( F ` x ) = ( G ` x ) } )
3		rabid2 2721	. . . 4 |- ( A = { x e. A | ( F ` x ) = ( G ` x ) } <-> A. x e. A ( F ` x ) = ( G ` x ) )
4	2, 3	bitri 184	. . 3 |- ( { x e. A | ( F ` x ) = ( G ` x ) } = A <-> A. x e. A ( F ` x ) = ( G ` x ) )
5	1, 4	bitr4di 198	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> { x e. A | ( F ` x ) = ( G ` x ) } = A ) )
6		fndmin 5785	. . 3 |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) = { x e. A | ( F ` x ) = ( G ` x ) } )
7	6	eqeq1d 2241	. 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 191	1 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> dom ( F i^i G ) = A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   A.wral 2520   {crab 2524   i^i cin 3210   dom cdm 4749   Fn wfn 5347   ` cfv 5352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360

This theorem is referenced by:  fneqeql2  5787  fnreseql  5788