Theorem dffv4g 5645

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 5112), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
dffv4g	|- ( A e. V -> ( F ` A ) = U. { x | ( F " { A } ) = { x } } )

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

Proof of Theorem dffv4g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffv3g 5644	. 2 |- ( A e. V -> ( F ` A ) = ( iota y y e. ( F " { A } ) ) )
2		df-iota 5293	. . 3 |- ( iota y y e. ( F " { A } ) ) = U. { x | { y | y e. ( F " { A } ) } = { x } }
3		abid2 2353	. . . . . 6 |- { y | y e. ( F " { A } ) } = ( F " { A } )
4	3	eqeq1i 2239	. . . . 5 |- ( { y | y e. ( F " { A } ) } = { x } <-> ( F " { A } ) = { x } )
5	4	abbii 2347	. . . 4 |- { x | { y | y e. ( F " { A } ) } = { x } } = { x | ( F " { A } ) = { x } }
6	5	unieqi 3908	. . 3 |- U. { x | { y | y e. ( F " { A } ) } = { x } } = U. { x | ( F " { A } ) = { x } }
7	2, 6	eqtri 2252	. 2 |- ( iota y y e. ( F " { A } ) ) = U. { x | ( F " { A } ) = { x } }
8	1, 7	eqtrdi 2280	1 |- ( A e. V -> ( F ` A ) = U. { x | ( F " { A } ) = { x } } )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   {cab 2217   {csn 3673   U.cuni 3898   "cima 4734   iotacio 5291   ` cfv 5333

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fv 5341

This theorem is referenced by: (None)