Theorem funconstss 5774

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 5705	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ { B } <-> A. x e. A ( F ` x ) e. { B } ) )
2		funimass3 5772	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ { B } <-> A C_ ( `' F " { B } ) ) )
3		ssel2 3223	. . . . . 6 |- ( ( A C_ dom F /\ x e. A ) -> x e. dom F )
4	3	anim2i 342	. . . . 5 |- ( ( Fun F /\ ( A C_ dom F /\ x e. A ) ) -> ( Fun F /\ x e. dom F ) )
5	4	anassrs 400	. . . 4 |- ( ( ( Fun F /\ A C_ dom F ) /\ x e. A ) -> ( Fun F /\ x e. dom F ) )
6		funfvex 5665	. . . 4 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
7		elsng 3688	. . . 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 2529	. 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 219	1 |- ( ( Fun F /\ A C_ dom F ) -> ( A. x e. A ( F ` x ) = B <-> A C_ ( `' F " { B } ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   C_ wss 3201   {csn 3673   `'ccnv 4730   dom cdm 4731   "cima 4734   Fun wfun 5327   ` cfv 5333

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by:  fconst3m  5881