Theorem fnsnfv 5738

Description: Singleton of function value. (Contributed by NM, 22-May-1998.)

Assertion

Ref	Expression
fnsnfv	|- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )

Proof of Theorem fnsnfv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcom 2236	. . . 4 |- ( y = ( F ` B ) <-> ( F ` B ) = y )
2		fnbrfvb 5717	. . . 4 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = y <-> B F y ) )
3	1, 2	bitrid 192	. . 3 |- ( ( F Fn A /\ B e. A ) -> ( y = ( F ` B ) <-> B F y ) )
4	3	abbidv 2354	. 2 |- ( ( F Fn A /\ B e. A ) -> { y | y = ( F ` B ) } = { y | B F y } )
5		df-sn 3697	. . 3 |- { ( F ` B ) } = { y | y = ( F ` B ) }
6	5	a1i 9	. 2 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = { y | y = ( F ` B ) } )
7		imasng 5129	. . 3 |- ( B e. A -> ( F " { B } ) = { y | B F y } )
8	7	adantl 277	. 2 |- ( ( F Fn A /\ B e. A ) -> ( F " { B } ) = { y | B F y } )
9	4, 6, 8	3eqtr4d 2277	1 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   {cab 2220   {csn 3691   class class class wbr 4111   "cima 4754   Fn wfn 5349   ` cfv 5354

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362

This theorem is referenced by:  fnimapr  5739  funfvdm  5742  fvco2  5748  fvimacnvi  5794  fsn2  5853  suppval1  6441  suppsnopdc  6452  phplem4  7111  phplem4dom  7118  phplem4on  7124  fiintim  7193  fidcenumlemrks  7225  fidcenumlemr  7227  resunimafz0  11202  ennnfonelemhf1o  13181