Theorem iotaexab 5237

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 4474	. 2 |- ( { x | ph } e. V -> U. { x | ph } e. _V )
2		abid 2184	. . . . 5 |- ( x e. { x | ph } <-> ph )
3		elssuni 3867	. . . . 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 1463	. . 3 |- A. x ( ph -> x C_ U. { x | ph } )
6		nfab1 2341	. . . . . . . 8 |- F/_ x { x | ph }
7	6	nfuni 3845	. . . . . . 7 |- F/_ x U. { x | ph }
8	7	nfeq2 2351	. . . . . 6 |- F/ x z = U. { x | ph }
9		sseq2 3207	. . . . . . 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 1629	. . . . 5 |- ( z = U. { x | ph } -> ( A. x ( ph -> x C_ z ) <-> A. x ( ph -> x C_ U. { x | ph } ) ) )
12		sseq2 3207	. . . . 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 5236	. . . 4 |- ( A. x ( ph -> x C_ z ) -> ( iota x ph ) C_ z )
15	13, 14	vtoclg 2824	. . 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 4173	1 |- ( { x | ph } e. V -> ( iota x ph ) e. _V )

Syntax hints:   -> wi 4   A.wal 1362   = wceq 1364   e. wcel 2167   {cab 2182   _Vcvv 2763   C_ wss 3157   U.cuni 3839   iotacio 5217

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-iota 5219

This theorem is referenced by:  fngsum  13031  igsumvalx  13032