Theorem iotaexab 5214

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 4457	. 2 |- ( { x | ph } e. V -> U. { x | ph } e. _V )
2		abid 2177	. . . . 5 |- ( x e. { x | ph } <-> ph )
3		elssuni 3852	. . . . 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 1460	. . 3 |- A. x ( ph -> x C_ U. { x | ph } )
6		nfab1 2334	. . . . . . . 8 |- F/_ x { x | ph }
7	6	nfuni 3830	. . . . . . 7 |- F/_ x U. { x | ph }
8	7	nfeq2 2344	. . . . . 6 |- F/ x z = U. { x | ph }
9		sseq2 3194	. . . . . . 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 1626	. . . . 5 |- ( z = U. { x | ph } -> ( A. x ( ph -> x C_ z ) <-> A. x ( ph -> x C_ U. { x | ph } ) ) )
12		sseq2 3194	. . . . 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 5213	. . . 4 |- ( A. x ( ph -> x C_ z ) -> ( iota x ph ) C_ z )
15	13, 14	vtoclg 2812	. . 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 1457	. 2 |- ( { x | ph } e. V -> ( iota x ph ) C_ U. { x | ph } )
17	1, 16	ssexd 4158	1 |- ( { x | ph } e. V -> ( iota x ph ) e. _V )

Syntax hints:   -> wi 4   A.wal 1362   = wceq 1364   e. wcel 2160   {cab 2175   _Vcvv 2752   C_ wss 3144   U.cuni 3824   iotacio 5194

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-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-iota 5196

This theorem is referenced by:  fngsum  12867  igsumvalx  12868