Theorem fneqeql 5637

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 5626	. . 3 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
2		eqcom 2189	. . . 4 |- ( { x e. A | ( F ` x ) = ( G ` x ) } = A <-> A = { x e. A | ( F ` x ) = ( G ` x ) } )
3		rabid2 2664	. . . 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 5636	. . 3 |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) = { x e. A | ( F ` x ) = ( G ` x ) } )
7	6	eqeq1d 2196	. 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 1363   A.wral 2465   {crab 2469   i^i cin 3140   dom cdm 4638   Fn wfn 5223   ` cfv 5228

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236

This theorem is referenced by:  fneqeql2  5638  fnreseql  5639