Theorem fneqeql2 5671

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 5670	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> dom ( F i^i G ) = A ) )
2		eqss 3198	. . 3 |- ( dom ( F i^i G ) = A <-> ( dom ( F i^i G ) C_ A /\ A C_ dom ( F i^i G ) ) )
3		inss1 3383	. . . . . 6 |- ( F i^i G ) C_ F
4		dmss 4865	. . . . . 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 5357	. . . . . 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 3233	. . . 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 1364   i^i cin 3156   C_ wss 3157   dom cdm 4663   Fn wfn 5253

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by: (None)