Theorem mptiniseg 5629

Description: Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Hypothesis

Ref	Expression
dmmpt.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
mptiniseg	|- ( C e. V -> ( `' F " { C } ) = { x e. A | B = C } )

Distinct variable groups:   x, C   x, V

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

Proof of Theorem mptiniseg

Step	Hyp	Ref	Expression
1		dmmpt.1	. . 3 |- F = ( x e. A |-> B )
2	1	mptpreima 5628	. 2 |- ( `' F " { C } ) = { x e. A | B e. { C } }
3		elsn2g 4210	. . 3 |- ( C e. V -> ( B e. { C } <-> B = C ) )
4	3	rabbidv 3189	. 2 |- ( C e. V -> { x e. A | B e. { C } } = { x e. A | B = C } )
5	2, 4	syl5eq 2668	1 |- ( C e. V -> ( `' F " { C } ) = { x e. A | B = C } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   {csn 4177   |-> cmpt 4729   `'ccnv 5113   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  ramub1lem1  15730  frlmsslss  20113  symgtgp  21905  csscld  23048  clsocv  23049  sqff1o  24908  dchrfi  24980  poimirlem30  33439  ftc1anclem6  33490  pwssplit4  37659  pwslnmlem2  37663