Theorem fnsnfv 5920

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 2471	. . . 4 |- ( y = ( F ` B ) <-> ( F ` B ) = y )
2		fnbrfvb 5901	. . . 4 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = y <-> B F y ) )
3	1, 2	syl5bb 257	. . 3 |- ( ( F Fn A /\ B e. A ) -> ( y = ( F ` B ) <-> B F y ) )
4	3	abbidv 2598	. 2 |- ( ( F Fn A /\ B e. A ) -> { y | y = ( F ` B ) } = { y | B F y } )
5		df-sn 4023	. . 3 |- { ( F ` B ) } = { y | y = ( F ` B ) }
6	5	a1i 11	. 2 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = { y | y = ( F ` B ) } )
7		fnrel 5672	. . . 4 |- ( F Fn A -> Rel F )
8		relimasn 5353	. . . 4 |- ( Rel F -> ( F " { B } ) = { y | B F y } )
9	7, 8	syl 16	. . 3 |- ( F Fn A -> ( F " { B } ) = { y | B F y } )
10	9	adantr 465	. 2 |- ( ( F Fn A /\ B e. A ) -> ( F " { B } ) = { y | B F y } )
11	4, 6, 10	3eqtr4d 2513	1 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1374   e. wcel 1762   {cab 2447   {csn 4022   class class class wbr 4442   "cima 4997   Rel wrel 4999   Fn wfn 5576   ` cfv 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-fv 5589

This theorem is referenced by:  fnimapr  5924  funfv  5927  fvco2  5935  fvimacnvi  5988  fvimacnvALT  5993  fsn2  6054  fparlem3  6877  fparlem4  6878  suppval1  6899  suppsnop  6907  domunsncan  7609  phplem4  7691  domunfican  7784  fiint  7788  infdifsn  8064  cantnfp1lem3  8090  cantnfp1lem3OLD  8116  symgfixelsi  16250  dprdf1o  16864  frlmlbs  18593  f1lindf  18619  cnt1  19612  xkohaus  19884  xkoptsub  19885  ustuqtop3  20476  2pthlem2  24262  eupath2lem3  24643  eulerpartlemmf  27942  grpokerinj  29939  funcoressn  31636