Theorem fneqeql2 5605

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 5604	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> dom ( F i^i G ) = A ) )
2		eqss 3162	. . 3 |- ( dom ( F i^i G ) = A <-> ( dom ( F i^i G ) C_ A /\ A C_ dom ( F i^i G ) ) )
3		inss1 3347	. . . . . 6 |- ( F i^i G ) C_ F
4		dmss 4810	. . . . . 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 5297	. . . . . 6 |- ( F Fn A -> dom F = A )
7	6	adantr 274	. . . . 5 |- ( ( F Fn A /\ G Fn A ) -> dom F = A )
8	5, 7	sseqtrid 3197	. . . 4 |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) C_ A )
9	8	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 ) ) ) )
10	2, 9	bitr4id 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 1348   i^i cin 3120   C_ wss 3121   dom cdm 4611   Fn wfn 5193

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by: (None)