Theorem fneqeql2 5744

Description: Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Assertion

Ref	Expression
fneqeql2	|- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A C_ dom ( F i^i G ) ) )

Proof of Theorem fneqeql2

Step	Hyp	Ref	Expression
1		fneqeql 5743	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> dom ( F i^i G ) = A ) )
2		eqss 3239	. . 3 |- ( dom ( F i^i G ) = A <-> ( dom ( F i^i G ) C_ A /\ A C_ dom ( F i^i G ) ) )
3		inss1 3424	. . . . . 6 |- ( F i^i G ) C_ F
4		dmss 4922	. . . . . 6 |- ( ( F i^i G ) C_ F -> dom ( F i^i G ) C_ dom F )
5	3, 4	ax-mp 5	. . . . 5 |- dom ( F i^i G ) C_ dom F
6		fndm 5420	. . . . . 6 |- ( F Fn A -> dom F = A )
7	6	adantr 276	. . . . 5 |- ( ( F Fn A /\ G Fn A ) -> dom F = A )
8	5, 7	sseqtrid 3274	. . . 4 |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) C_ A )
9	8	biantrurd 305	. . 3 |- ( ( F Fn A /\ G Fn A ) -> ( A C_ dom ( F i^i G ) <-> ( dom ( F i^i G ) C_ A /\ A C_ dom ( F i^i G ) ) ) )
10	2, 9	bitr4id 199	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( dom ( F i^i G ) = A <-> A C_ dom ( F i^i G ) ) )
11	1, 10	bitrd 188	1 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A C_ dom ( F i^i G ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   i^i cin 3196   C_ wss 3197   dom cdm 4719   Fn wfn 5313

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by: (None)