Theorem 0fv 5677

Description: Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)

Assertion

Ref	Expression
0fv	|- ( (/) ` A ) = (/)

Proof of Theorem 0fv

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 5334	. 2 |- ( (/) ` A ) = ( iota x A (/) x )
2		noel 3498	. . . . . 6 |- -. <. A , x >. e. (/)
3		df-br 4089	. . . . . 6 |- ( A (/) x <-> <. A , x >. e. (/) )
4	2, 3	mtbir 677	. . . . 5 |- -. A (/) x
5	4	nex 1548	. . . 4 |- -. E. x A (/) x
6		euex 2109	. . . 4 |- ( E! x A (/) x -> E. x A (/) x )
7	5, 6	mto 668	. . 3 |- -. E! x A (/) x
8		iotanul 5302	. . 3 |- ( -. E! x A (/) x -> ( iota x A (/) x ) = (/) )
9	7, 8	ax-mp 5	. 2 |- ( iota x A (/) x ) = (/)
10	1, 9	eqtri 2252	1 |- ( (/) ` A ) = (/)

Syntax hints:   -. wn 3   = wceq 1397   E.wex 1540   E!weu 2079   e. wcel 2202   (/)c0 3494   <.cop 3672   class class class wbr 4088   iotacio 5284   ` cfv 5326

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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334

This theorem is referenced by:  fv2prc  5678  ccat1st1st  11217  strsl0  13130