Theorem mptiniseg 5105

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 5104	. 2 |- ( `' F " { C } ) = { x e. A | B e. { C } }
3		elsn2g 3616	. . 3 |- ( C e. V -> ( B e. { C } <-> B = C ) )
4	3	rabbidv 2719	. 2 |- ( C e. V -> { x e. A | B e. { C } } = { x e. A | B = C } )
5	2, 4	eqtrid 2215	1 |- ( C e. V -> ( `' F " { C } ) = { x e. A | B = C } )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   {crab 2452   {csn 3583   |-> cmpt 4050   `'ccnv 4610   "cima 4614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-mpt 4052  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by: (None)