Theorem dffv4g 5555

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 5038), 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 5554	. 2 |- ( A e. V -> ( F ` A ) = ( iota y y e. ( F " { A } ) ) )
2		df-iota 5219	. . 3 |- ( iota y y e. ( F " { A } ) ) = U. { x | { y | y e. ( F " { A } ) } = { x } }
3		abid2 2317	. . . . . 6 |- { y | y e. ( F " { A } ) } = ( F " { A } )
4	3	eqeq1i 2204	. . . . 5 |- ( { y | y e. ( F " { A } ) } = { x } <-> ( F " { A } ) = { x } )
5	4	abbii 2312	. . . 4 |- { x | { y | y e. ( F " { A } ) } = { x } } = { x | ( F " { A } ) = { x } }
6	5	unieqi 3849	. . 3 |- U. { x | { y | y e. ( F " { A } ) } = { x } } = U. { x | ( F " { A } ) = { x } }
7	2, 6	eqtri 2217	. 2 |- ( iota y y e. ( F " { A } ) ) = U. { x | ( F " { A } ) = { x } }
8	1, 7	eqtrdi 2245	1 |- ( A e. V -> ( F ` A ) = U. { x | ( F " { A } ) = { x } } )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   {cab 2182   {csn 3622   U.cuni 3839   "cima 4666   iotacio 5217   ` cfv 5258

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fv 5266

This theorem is referenced by: (None)