Theorem iotaexab 5296

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 4529	. 2 |- ( { x | ph } e. V -> U. { x | ph } e. _V )
2		abid 2217	. . . . 5 |- ( x e. { x | ph } <-> ph )
3		elssuni 3915	. . . . 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 1495	. . 3 |- A. x ( ph -> x C_ U. { x | ph } )
6		nfab1 2374	. . . . . . . 8 |- F/_ x { x | ph }
7	6	nfuni 3893	. . . . . . 7 |- F/_ x U. { x | ph }
8	7	nfeq2 2384	. . . . . 6 |- F/ x z = U. { x | ph }
9		sseq2 3248	. . . . . . 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 1661	. . . . 5 |- ( z = U. { x | ph } -> ( A. x ( ph -> x C_ z ) <-> A. x ( ph -> x C_ U. { x | ph } ) ) )
12		sseq2 3248	. . . . 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 5295	. . . 4 |- ( A. x ( ph -> x C_ z ) -> ( iota x ph ) C_ z )
15	13, 14	vtoclg 2861	. . 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 1489	. 2 |- ( { x | ph } e. V -> ( iota x ph ) C_ U. { x | ph } )
17	1, 16	ssexd 4223	1 |- ( { x | ph } e. V -> ( iota x ph ) e. _V )

Syntax hints:   -> wi 4   A.wal 1393   = wceq 1395   e. wcel 2200   {cab 2215   _Vcvv 2799   C_ wss 3197   U.cuni 3887   iotacio 5275

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-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-iota 5277

This theorem is referenced by:  fngsum  13416  igsumvalx  13417