Theorem fconst7 39484

Description: An alternative way to express a constant function. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

Ref	Expression
fconst7.p	|- F/ x ph
fconst7.x	|- F/_ x F
fconst7.f	|- ( ph -> F Fn A )
fconst7.b	|- ( ph -> B e. V )
fconst7.e	|- ( ( ph /\ x e. A ) -> ( F ` x ) = B )

Assertion

Ref	Expression
fconst7	|- ( ph -> F = ( A X. { B } ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   F( x)   V( x)

Proof of Theorem fconst7

Step	Hyp	Ref	Expression
1		fconst7.f	. . 3 |- ( ph -> F Fn A )
2		fconst7.p	. . . 4 |- F/ x ph
3		fconst7.e	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
4		fvexd 6203	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. _V )
5	3, 4	eqeltrrd 2702	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. _V )
6		snidg 4206	. . . . . 6 |- ( B e. _V -> B e. { B } )
7	5, 6	syl 17	. . . . 5 |- ( ( ph /\ x e. A ) -> B e. { B } )
8	3, 7	eqeltrd 2701	. . . 4 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. { B } )
9	2, 8	ralrimia 39315	. . 3 |- ( ph -> A. x e. A ( F ` x ) e. { B } )
10		nfcv 2764	. . . 4 |- F/_ x A
11		nfcv 2764	. . . 4 |- F/_ x { B }
12		fconst7.x	. . . 4 |- F/_ x F
13	10, 11, 12	ffnfvf 6389	. . 3 |- ( F : A --> { B } <-> ( F Fn A /\ A. x e. A ( F ` x ) e. { B } ) )
14	1, 9, 13	sylanbrc 698	. 2 |- ( ph -> F : A --> { B } )
15		fconst7.b	. . 3 |- ( ph -> B e. V )
16		fconst2g 6468	. . 3 |- ( B e. V -> ( F : A --> { B } <-> F = ( A X. { B } ) ) )
17	15, 16	syl 17	. 2 |- ( ph -> ( F : A --> { B } <-> F = ( A X. { B } ) ) )
18	14, 17	mpbid 222	1 |- ( ph -> F = ( A X. { B } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   A.wral 2912   _Vcvv 3200   {csn 4177   X. cxp 5112   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-8 1992  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-pow 4843  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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  xlimconst  40051