Theorem funconstss 5765

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 5696	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ { B } <-> A. x e. A ( F ` x ) e. { B } ) )
2		funimass3 5763	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ { B } <-> A C_ ( `' F " { B } ) ) )
3		ssel2 3222	. . . . . 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 5656	. . . 4 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
7		elsng 3684	. . . 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 2528	. 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 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   {csn 3669   `'ccnv 4724   dom cdm 4725   "cima 4728   Fun wfun 5320   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by:  fconst3m  5872