Theorem iotaexab 5351

Description: Existence of the iota class when all the possible values are contained in a set. (Contributed by Jim Kingdon, 27-May-2025.)

Assertion

Ref	Expression
iotaexab	|- ( { x | ph } e. V -> ( iota x ph ) e. _V )

Proof of Theorem iotaexab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniexg 4580	. 2 |- ( { x | ph } e. V -> U. { x | ph } e. _V )
2		abid 2226	. . . . 5 |- ( x e. { x | ph } <-> ph )
3		elssuni 3958	. . . . 5 |- ( x e. { x | ph } -> x C_ U. { x | ph } )
4	2, 3	sylbir 135	. . . 4 |- ( ph -> x C_ U. { x | ph } )
5	4	ax-gen 1502	. . 3 |- A. x ( ph -> x C_ U. { x | ph } )
6		nfab1 2394	. . . . . . . 8 |- F/_ x { x | ph }
7	6	nfuni 3936	. . . . . . 7 |- F/_ x U. { x | ph }
8	7	nfeq2 2404	. . . . . 6 |- F/ x z = U. { x | ph }
9		sseq2 3272	. . . . . . 7 |- ( z = U. { x | ph } -> ( x C_ z <-> x C_ U. { x | ph } ) )
10	9	imbi2d 230	. . . . . 6 |- ( z = U. { x | ph } -> ( ( ph -> x C_ z ) <-> ( ph -> x C_ U. { x | ph } ) ) )
11	8, 10	albid 1668	. . . . 5 |- ( z = U. { x | ph } -> ( A. x ( ph -> x C_ z ) <-> A. x ( ph -> x C_ U. { x | ph } ) ) )
12		sseq2 3272	. . . . 5 |- ( z = U. { x | ph } -> ( ( iota x ph ) C_ z <-> ( iota x ph ) C_ U. { x | ph } ) )
13	11, 12	imbi12d 234	. . . 4 |- ( z = U. { x | ph } -> ( ( A. x ( ph -> x C_ z ) -> ( iota x ph ) C_ z ) <-> ( A. x ( ph -> x C_ U. { x | ph } ) -> ( iota x ph ) C_ U. { x | ph } ) ) )
14		iotass 5350	. . . 4 |- ( A. x ( ph -> x C_ z ) -> ( iota x ph ) C_ z )
15	13, 14	vtoclg 2883	. . 3 |- ( U. { x | ph } e. _V -> ( A. x ( ph -> x C_ U. { x | ph } ) -> ( iota x ph ) C_ U. { x | ph } ) )
16	1, 5, 15	mpisyl 1496	. 2 |- ( { x | ph } e. V -> ( iota x ph ) C_ U. { x | ph } )
17	1, 16	ssexd 4268	1 |- ( { x | ph } e. V -> ( iota x ph ) e. _V )

Syntax hints:   -> wi 4   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   C_ wss 3220   U.cuni 3930   iotacio 5330

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-un 4573

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-iota 5332

This theorem is referenced by:  fngzsum  13685  gzsumvalx  13686