Theorem mptiniseg 5231

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 5230	. 2 |- ( `' F " { C } ) = { x e. A | B e. { C } }
3		elsn2g 3702	. . 3 |- ( C e. V -> ( B e. { C } <-> B = C ) )
4	3	rabbidv 2791	. 2 |- ( C e. V -> { x e. A | B e. { C } } = { x e. A | B = C } )
5	2, 4	eqtrid 2276	1 |- ( C e. V -> ( `' F " { C } ) = { x e. A | B = C } )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   {crab 2514   {csn 3669   |-> cmpt 4150   `'ccnv 4724   "cima 4728

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-mpt 4152  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by: (None)