Theorem fconst3m 5380

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 5379	. 2 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A. y e. A ( F ` y ) = B ) ) )
2		fnfun 4996	. . . 4 |- ( F Fn A -> Fun F )
3		fndm 4998	. . . . 5 |- ( F Fn A -> dom F = A )
4		eqimss2 2998	. . . . 5 |- ( dom F = A -> A C_ dom F )
5	3, 4	syl 14	. . . 4 |- ( F Fn A -> A C_ dom F )
6		funconstss 5285	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( A. y e. A ( F ` y ) = B <-> A C_ ( `' F " { B } ) ) )
7	2, 5, 6	syl2anc 391	. . 3 |- ( F Fn A -> ( A. y e. A ( F ` y ) = B <-> A C_ ( `' F " { B } ) ) )
8	7	pm5.32i 427	. 2 |- ( ( F Fn A /\ A. y e. A ( F ` y ) = B ) <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) )
9	1, 8	syl6bb 185	1 |- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   A.wral 2306   C_ wss 2917   {csn 3375   `'ccnv 4344   dom cdm 4345   "cima 4348   Fun wfun 4896   Fn wfn 4897   -->wf 4898   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910

This theorem is referenced by:  fconst4m  5381