Theorem dffv4g 5624

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 5097), 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 5623	. 2 |- ( A e. V -> ( F ` A ) = ( iota y y e. ( F " { A } ) ) )
2		df-iota 5278	. . 3 |- ( iota y y e. ( F " { A } ) ) = U. { x | { y | y e. ( F " { A } ) } = { x } }
3		abid2 2350	. . . . . 6 |- { y | y e. ( F " { A } ) } = ( F " { A } )
4	3	eqeq1i 2237	. . . . 5 |- ( { y | y e. ( F " { A } ) } = { x } <-> ( F " { A } ) = { x } )
5	4	abbii 2345	. . . 4 |- { x | { y | y e. ( F " { A } ) } = { x } } = { x | ( F " { A } ) = { x } }
6	5	unieqi 3898	. . 3 |- U. { x | { y | y e. ( F " { A } ) } = { x } } = U. { x | ( F " { A } ) = { x } }
7	2, 6	eqtri 2250	. 2 |- ( iota y y e. ( F " { A } ) ) = U. { x | ( F " { A } ) = { x } }
8	1, 7	eqtrdi 2278	1 |- ( A e. V -> ( F ` A ) = U. { x | ( F " { A } ) = { x } } )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {cab 2215   {csn 3666   U.cuni 3888   "cima 4722   iotacio 5276   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fv 5326

This theorem is referenced by: (None)