Theorem fneqeql 5296

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 5286	. . 3 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
2		eqcom 2083	. . . 4 |- ( { x e. A | ( F ` x ) = ( G ` x ) } = A <-> A = { x e. A | ( F ` x ) = ( G ` x ) } )
3		rabid2 2530	. . . 4 |- ( A = { x e. A | ( F ` x ) = ( G ` x ) } <-> A. x e. A ( F ` x ) = ( G ` x ) )
4	2, 3	bitri 182	. . 3 |- ( { x e. A | ( F ` x ) = ( G ` x ) } = A <-> A. x e. A ( F ` x ) = ( G ` x ) )
5	1, 4	syl6bbr 196	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> { x e. A | ( F ` x ) = ( G ` x ) } = A ) )
6		fndmin 5295	. . 3 |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) = { x e. A | ( F ` x ) = ( G ` x ) } )
7	6	eqeq1d 2089	. 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 189	1 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> dom ( F i^i G ) = A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   A.wral 2348   {crab 2352   i^i cin 2972   dom cdm 4363   Fn wfn 4917   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by:  fneqeql2  5297  fnreseql  5298