Theorem fconst3m 5777

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 5776	. 2 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A. y e. A ( F ` y ) = B ) ) )
2		fnfun 5351	. . . 4 |- ( F Fn A -> Fun F )
3		fndm 5353	. . . . 5 |- ( F Fn A -> dom F = A )
4		eqimss2 3234	. . . . 5 |- ( dom F = A -> A C_ dom F )
5	3, 4	syl 14	. . . 4 |- ( F Fn A -> A C_ dom F )
6		funconstss 5676	. . . 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 2164   A.wral 2472   C_ wss 3153   {csn 3618   `'ccnv 4658   dom cdm 4659   "cima 4662   Fun wfun 5248   Fn wfn 5249   -->wf 5250   ` cfv 5254

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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260  df-fv 5262

This theorem is referenced by:  fconst4m  5778