Theorem dffv3 6187

Description: A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)

Assertion

Ref	Expression
dffv3	|- ( F ` A ) = ( iota x x e. ( F " { A } ) )

Distinct variable groups:   x, F   x, A

Proof of Theorem dffv3

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- x e. _V
2		elimasng 5491	. . . . . 6 |- ( ( A e. _V /\ x e. _V ) -> ( x e. ( F " { A } ) <-> <. A , x >. e. F ) )
3		df-br 4654	. . . . . 6 |- ( A F x <-> <. A , x >. e. F )
4	2, 3	syl6bbr 278	. . . . 5 |- ( ( A e. _V /\ x e. _V ) -> ( x e. ( F " { A } ) <-> A F x ) )
5	1, 4	mpan2 707	. . . 4 |- ( A e. _V -> ( x e. ( F " { A } ) <-> A F x ) )
6	5	iotabidv 5872	. . 3 |- ( A e. _V -> ( iota x x e. ( F " { A } ) ) = ( iota x A F x ) )
7		df-fv 5896	. . 3 |- ( F ` A ) = ( iota x A F x )
8	6, 7	syl6reqr 2675	. 2 |- ( A e. _V -> ( F ` A ) = ( iota x x e. ( F " { A } ) ) )
9		fvprc 6185	. . 3 |- ( -. A e. _V -> ( F ` A ) = (/) )
10		snprc 4253	. . . . . . . . 9 |- ( -. A e. _V <-> { A } = (/) )
11	10	biimpi 206	. . . . . . . 8 |- ( -. A e. _V -> { A } = (/) )
12	11	imaeq2d 5466	. . . . . . 7 |- ( -. A e. _V -> ( F " { A } ) = ( F " (/) ) )
13		ima0 5481	. . . . . . 7 |- ( F " (/) ) = (/)
14	12, 13	syl6eq 2672	. . . . . 6 |- ( -. A e. _V -> ( F " { A } ) = (/) )
15	14	eleq2d 2687	. . . . 5 |- ( -. A e. _V -> ( x e. ( F " { A } ) <-> x e. (/) ) )
16	15	iotabidv 5872	. . . 4 |- ( -. A e. _V -> ( iota x x e. ( F " { A } ) ) = ( iota x x e. (/) ) )
17		noel 3919	. . . . . . 7 |- -. x e. (/)
18	17	nex 1731	. . . . . 6 |- -. E. x x e. (/)
19		euex 2494	. . . . . 6 |- ( E! x x e. (/) -> E. x x e. (/) )
20	18, 19	mto 188	. . . . 5 |- -. E! x x e. (/)
21		iotanul 5866	. . . . 5 |- ( -. E! x x e. (/) -> ( iota x x e. (/) ) = (/) )
22	20, 21	ax-mp 5	. . . 4 |- ( iota x x e. (/) ) = (/)
23	16, 22	syl6eq 2672	. . 3 |- ( -. A e. _V -> ( iota x x e. ( F " { A } ) ) = (/) )
24	9, 23	eqtr4d 2659	. 2 |- ( -. A e. _V -> ( F ` A ) = ( iota x x e. ( F " { A } ) ) )
25	8, 24	pm2.61i 176	1 |- ( F ` A ) = ( iota x x e. ( F " { A } ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   "cima 5117   iotacio 5849   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896

This theorem is referenced by:  dffv4  6188  fvco2  6273  shftval  13814  dffv5  32031