Theorem mptiniseg 5178

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 5177	. 2 |- ( `' F " { C } ) = { x e. A | B e. { C } }
3		elsn2g 3666	. . 3 |- ( C e. V -> ( B e. { C } <-> B = C ) )
4	3	rabbidv 2761	. 2 |- ( C e. V -> { x e. A | B e. { C } } = { x e. A | B = C } )
5	2, 4	eqtrid 2250	1 |- ( C e. V -> ( `' F " { C } ) = { x e. A | B = C } )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   {crab 2488   {csn 3633   |-> cmpt 4106   `'ccnv 4675   "cima 4679

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-mpt 4108  df-xp 4682  df-rel 4683  df-cnv 4684  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689

This theorem is referenced by: (None)