Theorem fconst3m 5401

Description: Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)

Assertion

Ref	Expression
fconst3m	|- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) ) )

Distinct variable group:   x, A

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

Proof of Theorem fconst3m

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconstfvm 5400	. 2 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A. y e. A ( F ` y ) = B ) ) )
2		fnfun 5016	. . . 4 |- ( F Fn A -> Fun F )
3		fndm 5018	. . . . 5 |- ( F Fn A -> dom F = A )
4		eqimss2 3052	. . . . 5 |- ( dom F = A -> A C_ dom F )
5	3, 4	syl 14	. . . 4 |- ( F Fn A -> A C_ dom F )
6		funconstss 5306	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( A. y e. A ( F ` y ) = B <-> A C_ ( `' F " { B } ) ) )
7	2, 5, 6	syl2anc 403	. . 3 |- ( F Fn A -> ( A. y e. A ( F ` y ) = B <-> A C_ ( `' F " { B } ) ) )
8	7	pm5.32i 441	. 2 |- ( ( F Fn A /\ A. y e. A ( F ` y ) = B ) <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) )
9	1, 8	syl6bb 194	1 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   A.wral 2348   C_ wss 2973   {csn 3398   `'ccnv 4362   dom cdm 4363   "cima 4366   Fun wfun 4916   Fn wfn 4917   -->wf 4918   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930

This theorem is referenced by:  fconst4m  5402