Theorem fneqeql2 5483

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 5482	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> dom ( F i^i G ) = A ) )
2		inss1 3262	. . . . . 6 |- ( F i^i G ) C_ F
3		dmss 4698	. . . . . 6 |- ( ( F i^i G ) C_ F -> dom ( F i^i G ) C_ dom F )
4	2, 3	ax-mp 7	. . . . 5 |- dom ( F i^i G ) C_ dom F
5		fndm 5180	. . . . . 6 |- ( F Fn A -> dom F = A )
6	5	adantr 272	. . . . 5 |- ( ( F Fn A /\ G Fn A ) -> dom F = A )
7	4, 6	sseqtrid 3113	. . . 4 |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) C_ A )
8	7	biantrurd 301	. . 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 3078	. . 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 1314   i^i cin 3036   C_ wss 3037   dom cdm 4499   Fn wfn 5076

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fn 5084  df-fv 5089

This theorem is referenced by: (None)