Theorem dffv4 6188

Description: The previous definition of function value, from before the iota operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 5493), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
dffv4	|- ( F ` A ) = U. { x | ( F " { A } ) = { x } }

Distinct variable groups:   x, A   x, F

Proof of Theorem dffv4

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffv3 6187	. 2 |- ( F ` A ) = ( iota y y e. ( F " { A } ) )
2		df-iota 5851	. 2 |- ( iota y y e. ( F " { A } ) ) = U. { x | { y | y e. ( F " { A } ) } = { x } }
3		abid2 2745	. . . . 5 |- { y | y e. ( F " { A } ) } = ( F " { A } )
4	3	eqeq1i 2627	. . . 4 |- ( { y | y e. ( F " { A } ) } = { x } <-> ( F " { A } ) = { x } )
5	4	abbii 2739	. . 3 |- { x | { y | y e. ( F " { A } ) } = { x } } = { x | ( F " { A } ) = { x } }
6	5	unieqi 4445	. 2 |- U. { x | { y | y e. ( F " { A } ) } = { x } } = U. { x | ( F " { A } ) = { x } }
7	1, 2, 6	3eqtri 2648	1 |- ( F ` A ) = U. { x | ( F " { A } ) = { x } }

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   {csn 4177   U.cuni 4436   "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:  csbfv12gALTOLD  39052  csbfv12gALTVD  39135