Theorem fconstfvm 5880

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5879. (Contributed by Jim Kingdon, 8-Jan-2019.)

Assertion

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

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

Allowed substitution hints:   B( y)   F( y)

Proof of Theorem fconstfvm

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 5489	. . 3 |- ( F : A --> { B } -> F Fn A )
2		fvconst 5850	. . . 4 |- ( ( F : A --> { B } /\ x e. A ) -> ( F ` x ) = B )
3	2	ralrimiva 2606	. . 3 |- ( F : A --> { B } -> A. x e. A ( F ` x ) = B )
4	1, 3	jca 306	. 2 |- ( F : A --> { B } -> ( F Fn A /\ A. x e. A ( F ` x ) = B ) )
5		fvelrnb 5702	. . . . . . . . 9 |- ( F Fn A -> ( w e. ran F <-> E. z e. A ( F ` z ) = w ) )
6		fveq2 5648	. . . . . . . . . . . . . 14 |- ( x = z -> ( F ` x ) = ( F ` z ) )
7	6	eqeq1d 2240	. . . . . . . . . . . . 13 |- ( x = z -> ( ( F ` x ) = B <-> ( F ` z ) = B ) )
8	7	rspccva 2910	. . . . . . . . . . . 12 |- ( ( A. x e. A ( F ` x ) = B /\ z e. A ) -> ( F ` z ) = B )
9	8	eqeq1d 2240	. . . . . . . . . . 11 |- ( ( A. x e. A ( F ` x ) = B /\ z e. A ) -> ( ( F ` z ) = w <-> B = w ) )
10	9	rexbidva 2530	. . . . . . . . . 10 |- ( A. x e. A ( F ` x ) = B -> ( E. z e. A ( F ` z ) = w <-> E. z e. A B = w ) )
11		r19.9rmv 3588	. . . . . . . . . . 11 |- ( E. y y e. A -> ( B = w <-> E. z e. A B = w ) )
12	11	bicomd 141	. . . . . . . . . 10 |- ( E. y y e. A -> ( E. z e. A B = w <-> B = w ) )
13	10, 12	sylan9bbr 463	. . . . . . . . 9 |- ( ( E. y y e. A /\ A. x e. A ( F ` x ) = B ) -> ( E. z e. A ( F ` z ) = w <-> B = w ) )
14	5, 13	sylan9bbr 463	. . . . . . . 8 |- ( ( ( E. y y e. A /\ A. x e. A ( F ` x ) = B ) /\ F Fn A ) -> ( w e. ran F <-> B = w ) )
15		velsn 3690	. . . . . . . . 9 |- ( w e. { B } <-> w = B )
16		eqcom 2233	. . . . . . . . 9 |- ( w = B <-> B = w )
17	15, 16	bitr2i 185	. . . . . . . 8 |- ( B = w <-> w e. { B } )
18	14, 17	bitrdi 196	. . . . . . 7 |- ( ( ( E. y y e. A /\ A. x e. A ( F ` x ) = B ) /\ F Fn A ) -> ( w e. ran F <-> w e. { B } ) )
19	18	eqrdv 2229	. . . . . 6 |- ( ( ( E. y y e. A /\ A. x e. A ( F ` x ) = B ) /\ F Fn A ) -> ran F = { B } )
20	19	an32s 570	. . . . 5 |- ( ( ( E. y y e. A /\ F Fn A ) /\ A. x e. A ( F ` x ) = B ) -> ran F = { B } )
21	20	exp31 364	. . . 4 |- ( E. y y e. A -> ( F Fn A -> ( A. x e. A ( F ` x ) = B -> ran F = { B } ) ) )
22	21	imdistand 447	. . 3 |- ( E. y y e. A -> ( ( F Fn A /\ A. x e. A ( F ` x ) = B ) -> ( F Fn A /\ ran F = { B } ) ) )
23		df-fo 5339	. . . 4 |- ( F : A -onto-> { B } <-> ( F Fn A /\ ran F = { B } ) )
24		fof 5568	. . . 4 |- ( F : A -onto-> { B } -> F : A --> { B } )
25	23, 24	sylbir 135	. . 3 |- ( ( F Fn A /\ ran F = { B } ) -> F : A --> { B } )
26	22, 25	syl6 33	. 2 |- ( E. y y e. A -> ( ( F Fn A /\ A. x e. A ( F ` x ) = B ) -> F : A --> { B } ) )
27	4, 26	impbid2 143	1 |- ( E. y y e. A -> ( F : A --> { B } <-> ( F Fn A /\ A. x e. A ( F ` x ) = B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2202   A.wral 2511   E.wrex 2512   {csn 3673   ran crn 4732   Fn wfn 5328   -->wf 5329   -onto->wfo 5331   ` 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-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341

This theorem is referenced by:  fconst3m  5881