Theorem funconstss 5711

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 5642	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ { B } <-> A. x e. A ( F ` x ) e. { B } ) )
2		funimass3 5709	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ { B } <-> A C_ ( `' F " { B } ) ) )
3		ssel2 3192	. . . . . 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 5606	. . . 4 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
7		elsng 3653	. . . 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 2503	. 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 1373   e. wcel 2177   A.wral 2485   _Vcvv 2773   C_ wss 3170   {csn 3638   `'ccnv 4682   dom cdm 4683   "cima 4686   Fun wfun 5274   ` cfv 5280

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-fv 5288

This theorem is referenced by:  fconst3m  5816