Theorem dffv4g 5687

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 5151), 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 5686	. 2 |- ( A e. V -> ( F ` A ) = ( iota y y e. ( F " { A } ) ) )
2		df-iota 5332	. . 3 |- ( iota y y e. ( F " { A } ) ) = U. { x | { y | y e. ( F " { A } ) } = { x } }
3		abid2 2361	. . . . . 6 |- { y | y e. ( F " { A } ) } = ( F " { A } )
4	3	eqeq1i 2246	. . . . 5 |- ( { y | y e. ( F " { A } ) } = { x } <-> ( F " { A } ) = { x } )
5	4	abbii 2354	. . . 4 |- { x | { y | y e. ( F " { A } ) } = { x } } = { x | ( F " { A } ) = { x } }
6	5	unieqi 3940	. . 3 |- U. { x | { y | y e. ( F " { A } ) } = { x } } = U. { x | ( F " { A } ) = { x } }
7	2, 6	eqtri 2259	. 2 |- ( iota y y e. ( F " { A } ) ) = U. { x | ( F " { A } ) = { x } }
8	1, 7	eqtrdi 2287	1 |- ( A e. V -> ( F ` A ) = U. { x | ( F " { A } ) = { x } } )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {cab 2224   {csn 3705   U.cuni 3930   "cima 4772   iotacio 5330   ` cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fv 5380

This theorem is referenced by: (None)