Theorem fconstfv 6476

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 6470. (Contributed by NM, 27-Aug-2004.) (Proof shortened by OpenAI, 25-Mar-2020.)

Assertion

Ref	Expression
fconstfv	|- ( F : A --> { B } <-> ( F Fn A /\ A. x e. A ( F ` x ) = B ) )

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

Proof of Theorem fconstfv

Step	Hyp	Ref	Expression
1		ffnfv 6388	. 2 |- ( F : A --> { B } <-> ( F Fn A /\ A. x e. A ( F ` x ) e. { B } ) )
2		fvex 6201	. . . . 5 |- ( F ` x ) e. _V
3	2	elsn 4192	. . . 4 |- ( ( F ` x ) e. { B } <-> ( F ` x ) = B )
4	3	ralbii 2980	. . 3 |- ( A. x e. A ( F ` x ) e. { B } <-> A. x e. A ( F ` x ) = B )
5	4	anbi2i 730	. 2 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. { B } ) <-> ( F Fn A /\ A. x e. A ( F ` x ) = B ) )
6	1, 5	bitri 264	1 |- ( F : A --> { B } <-> ( F Fn A /\ A. x e. A ( F ` x ) = B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {csn 4177   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:  fconst3  6477  repsdf2  13525  rrxcph  23180  lnon0  27653  df0op2  28611  matunitlindflem1  33405  poimir  33442  lfl1  34357