Theorem frnssb 6391

Description: A function is a function into a subset of its codomain if all of its values are elements of this subset. (Contributed by AV, 7-Feb-2021.)

Assertion

Ref	Expression
frnssb	|- ( ( V C_ W /\ A. k e. A ( F ` k ) e. V ) -> ( F : A --> W <-> F : A --> V ) )

Distinct variable groups:   A, k   k, F   k, V

Allowed substitution hint:   W( k)

Proof of Theorem frnssb

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( V C_ W /\ A. k e. A ( F ` k ) e. V ) -> A. k e. A ( F ` k ) e. V )
2		ffn 6045	. . . 4 |- ( F : A --> W -> F Fn A )
3	1, 2	anim12ci 591	. . 3 |- ( ( ( V C_ W /\ A. k e. A ( F ` k ) e. V ) /\ F : A --> W ) -> ( F Fn A /\ A. k e. A ( F ` k ) e. V ) )
4		ffnfv 6388	. . 3 |- ( F : A --> V <-> ( F Fn A /\ A. k e. A ( F ` k ) e. V ) )
5	3, 4	sylibr 224	. 2 |- ( ( ( V C_ W /\ A. k e. A ( F ` k ) e. V ) /\ F : A --> W ) -> F : A --> V )
6		simpl 473	. . . . 5 |- ( ( V C_ W /\ A. k e. A ( F ` k ) e. V ) -> V C_ W )
7	6	anim1i 592	. . . 4 |- ( ( ( V C_ W /\ A. k e. A ( F ` k ) e. V ) /\ F : A --> V ) -> ( V C_ W /\ F : A --> V ) )
8	7	ancomd 467	. . 3 |- ( ( ( V C_ W /\ A. k e. A ( F ` k ) e. V ) /\ F : A --> V ) -> ( F : A --> V /\ V C_ W ) )
9		fss 6056	. . 3 |- ( ( F : A --> V /\ V C_ W ) -> F : A --> W )
10	8, 9	syl 17	. 2 |- ( ( ( V C_ W /\ A. k e. A ( F ` k ) e. V ) /\ F : A --> V ) -> F : A --> W )
11	5, 10	impbida 877	1 |- ( ( V C_ W /\ A. k e. A ( F ` k ) e. V ) -> ( F : A --> W <-> F : A --> V ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   C_ wss 3574   Fn wfn 5883   -->wf 5884   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  wlkdlem1  26579