Theorem fneqeql2 5522

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 5521	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> dom ( F i^i G ) = A ) )
2		inss1 3291	. . . . . 6 |- ( F i^i G ) C_ F
3		dmss 4733	. . . . . 6 |- ( ( F i^i G ) C_ F -> dom ( F i^i G ) C_ dom F )
4	2, 3	ax-mp 5	. . . . 5 |- dom ( F i^i G ) C_ dom F
5		fndm 5217	. . . . . 6 |- ( F Fn A -> dom F = A )
6	5	adantr 274	. . . . 5 |- ( ( F Fn A /\ G Fn A ) -> dom F = A )
7	4, 6	sseqtrid 3142	. . . 4 |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) C_ A )
8	7	biantrurd 303	. . 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 ) ) ) )
9		eqss 3107	. . 3 |- ( dom ( F i^i G ) = A <-> ( dom ( F i^i G ) C_ A /\ A C_ dom ( F i^i G ) ) )
10	8, 9	syl6rbbr 198	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( dom ( F i^i G ) = A <-> A C_ dom ( F i^i G ) ) )
11	1, 10	bitrd 187	1 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A C_ dom ( F i^i G ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   i^i cin 3065   C_ wss 3066   dom cdm 4534   Fn wfn 5113

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by: (None)