Theorem fconst3m 5704

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 5703	. 2 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A. y e. A ( F ` y ) = B ) ) )
2		fnfun 5285	. . . 4 |- ( F Fn A -> Fun F )
3		fndm 5287	. . . . 5 |- ( F Fn A -> dom F = A )
4		eqimss2 3197	. . . . 5 |- ( dom F = A -> A C_ dom F )
5	3, 4	syl 14	. . . 4 |- ( F Fn A -> A C_ dom F )
6		funconstss 5603	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( A. y e. A ( F ` y ) = B <-> A C_ ( `' F " { B } ) ) )
7	2, 5, 6	syl2anc 409	. . 3 |- ( F Fn A -> ( A. y e. A ( F ` y ) = B <-> A C_ ( `' F " { B } ) ) )
8	7	pm5.32i 450	. 2 |- ( ( F Fn A /\ A. y e. A ( F ` y ) = B ) <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) )
9	1, 8	bitrdi 195	1 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   A.wral 2444   C_ wss 3116   {csn 3576   `'ccnv 4603   dom cdm 4604   "cima 4607   Fun wfun 5182   Fn wfn 5183   -->wf 5184   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196

This theorem is referenced by:  fconst4m  5705