Theorem dffv3g 5666

Description: A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)

Assertion

Ref	Expression
dffv3g	|- ( A e. V -> ( F ` A ) = ( iota x x e. ( F " { A } ) ) )

Distinct variable groups:   x, F   x, A   x, V

Proof of Theorem dffv3g

Step	Hyp	Ref	Expression
1		df-fv 5360	. 2 |- ( F ` A ) = ( iota x A F x )
2		vex 2816	. . . 4 |- x e. _V
3		elimasng 5130	. . . . 5 |- ( ( A e. V /\ x e. _V ) -> ( x e. ( F " { A } ) <-> <. A , x >. e. F ) )
4		df-br 4110	. . . . 5 |- ( A F x <-> <. A , x >. e. F )
5	3, 4	bitr4di 198	. . . 4 |- ( ( A e. V /\ x e. _V ) -> ( x e. ( F " { A } ) <-> A F x ) )
6	2, 5	mpan2 425	. . 3 |- ( A e. V -> ( x e. ( F " { A } ) <-> A F x ) )
7	6	iotabidv 5335	. 2 |- ( A e. V -> ( iota x x e. ( F " { A } ) ) = ( iota x A F x ) )
8	1, 7	eqtr4id 2284	1 |- ( A e. V -> ( F ` A ) = ( iota x x e. ( F " { A } ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   _Vcvv 2813   {csn 3689   <.cop 3692   class class class wbr 4109   "cima 4752   iotacio 5310   ` cfv 5352

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fv 5360

This theorem is referenced by:  dffv4g  5667  fvco2  5746  shftval  11510