Theorem fconst3m 5751

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 5750	. 2 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A. y e. A ( F ` y ) = B ) ) )
2		fnfun 5328	. . . 4 |- ( F Fn A -> Fun F )
3		fndm 5330	. . . . 5 |- ( F Fn A -> dom F = A )
4		eqimss2 3225	. . . . 5 |- ( dom F = A -> A C_ dom F )
5	3, 4	syl 14	. . . 4 |- ( F Fn A -> A C_ dom F )
6		funconstss 5650	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( A. y e. A ( F ` y ) = B <-> A C_ ( `' F " { B } ) ) )
7	2, 5, 6	syl2anc 411	. . 3 |- ( F Fn A -> ( A. y e. A ( F ` y ) = B <-> A C_ ( `' F " { B } ) ) )
8	7	pm5.32i 454	. 2 |- ( ( F Fn A /\ A. y e. A ( F ` y ) = B ) <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) )
9	1, 8	bitrdi 196	1 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2160   A.wral 2468   C_ wss 3144   {csn 3607   `'ccnv 4640   dom cdm 4641   "cima 4644   Fun wfun 5225   Fn wfn 5226   -->wf 5227   ` cfv 5231

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fo 5237  df-fv 5239

This theorem is referenced by:  fconst4m  5752