Theorem funconstss 5603

Description: Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)

Assertion

Ref	Expression
funconstss	|- ( ( Fun F /\ A C_ dom F ) -> ( A. x e. A ( F ` x ) = B <-> A C_ ( `' F " { B } ) ) )

Distinct variable groups:   x, F   x, A   x, B

Proof of Theorem funconstss

Step	Hyp	Ref	Expression
1		funimass4 5537	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ { B } <-> A. x e. A ( F ` x ) e. { B } ) )
2		funimass3 5601	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ { B } <-> A C_ ( `' F " { B } ) ) )
3		ssel2 3137	. . . . . 6 |- ( ( A C_ dom F /\ x e. A ) -> x e. dom F )
4	3	anim2i 340	. . . . 5 |- ( ( Fun F /\ ( A C_ dom F /\ x e. A ) ) -> ( Fun F /\ x e. dom F ) )
5	4	anassrs 398	. . . 4 |- ( ( ( Fun F /\ A C_ dom F ) /\ x e. A ) -> ( Fun F /\ x e. dom F ) )
6		funfvex 5503	. . . 4 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
7		elsng 3591	. . . 4 |- ( ( F ` x ) e. _V -> ( ( F ` x ) e. { B } <-> ( F ` x ) = B ) )
8	5, 6, 7	3syl 17	. . 3 |- ( ( ( Fun F /\ A C_ dom F ) /\ x e. A ) -> ( ( F ` x ) e. { B } <-> ( F ` x ) = B ) )
9	8	ralbidva 2462	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( A. x e. A ( F ` x ) e. { B } <-> A. x e. A ( F ` x ) = B ) )
10	1, 2, 9	3bitr3rd 218	1 |- ( ( Fun F /\ A C_ dom F ) -> ( A. x e. A ( F ` x ) = B <-> A C_ ( `' F " { B } ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   _Vcvv 2726   C_ wss 3116   {csn 3576   `'ccnv 4603   dom cdm 4604   "cima 4607   Fun wfun 5182   ` cfv 5188

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by:  fconst3m  5704