Theorem mptiniseg 5115

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

Hypothesis

Ref	Expression
dmmpo.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		dmmpo.1	. . 3 |- F = ( x e. A |-> B )
2	1	mptpreima 5114	. 2 |- ( `' F " { C } ) = { x e. A | B e. { C } }
3		elsn2g 3622	. . 3 |- ( C e. V -> ( B e. { C } <-> B = C ) )
4	3	rabbidv 2724	. 2 |- ( C e. V -> { x e. A | B e. { C } } = { x e. A | B = C } )
5	2, 4	eqtrid 2220	1 |- ( C e. V -> ( `' F " { C } ) = { x e. A | B = C } )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2146   {crab 2457   {csn 3589   |-> cmpt 4059   `'ccnv 4619   "cima 4623

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-mpt 4061  df-xp 4626  df-rel 4627  df-cnv 4628  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633

This theorem is referenced by: (None)