Theorem dffv4g 5551

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 5034), 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 5550	. 2 |- ( A e. V -> ( F ` A ) = ( iota y y e. ( F " { A } ) ) )
2		df-iota 5215	. . 3 |- ( iota y y e. ( F " { A } ) ) = U. { x | { y | y e. ( F " { A } ) } = { x } }
3		abid2 2314	. . . . . 6 |- { y | y e. ( F " { A } ) } = ( F " { A } )
4	3	eqeq1i 2201	. . . . 5 |- ( { y | y e. ( F " { A } ) } = { x } <-> ( F " { A } ) = { x } )
5	4	abbii 2309	. . . 4 |- { x | { y | y e. ( F " { A } ) } = { x } } = { x | ( F " { A } ) = { x } }
6	5	unieqi 3845	. . 3 |- U. { x | { y | y e. ( F " { A } ) } = { x } } = U. { x | ( F " { A } ) = { x } }
7	2, 6	eqtri 2214	. 2 |- ( iota y y e. ( F " { A } ) ) = U. { x | ( F " { A } ) = { x } }
8	1, 7	eqtrdi 2242	1 |- ( A e. V -> ( F ` A ) = U. { x | ( F " { A } ) = { x } } )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   {cab 2179   {csn 3618   U.cuni 3835   "cima 4662   iotacio 5213   ` cfv 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fv 5262

This theorem is referenced by: (None)