Theorem iotaexab 5264

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 4499	. 2 |- ( { x | ph } e. V -> U. { x | ph } e. _V )
2		abid 2194	. . . . 5 |- ( x e. { x | ph } <-> ph )
3		elssuni 3887	. . . . 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 1473	. . 3 |- A. x ( ph -> x C_ U. { x | ph } )
6		nfab1 2351	. . . . . . . 8 |- F/_ x { x | ph }
7	6	nfuni 3865	. . . . . . 7 |- F/_ x U. { x | ph }
8	7	nfeq2 2361	. . . . . 6 |- F/ x z = U. { x | ph }
9		sseq2 3221	. . . . . . 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 1639	. . . . 5 |- ( z = U. { x | ph } -> ( A. x ( ph -> x C_ z ) <-> A. x ( ph -> x C_ U. { x | ph } ) ) )
12		sseq2 3221	. . . . 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 5263	. . . 4 |- ( A. x ( ph -> x C_ z ) -> ( iota x ph ) C_ z )
15	13, 14	vtoclg 2835	. . 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 1467	. 2 |- ( { x | ph } e. V -> ( iota x ph ) C_ U. { x | ph } )
17	1, 16	ssexd 4195	1 |- ( { x | ph } e. V -> ( iota x ph ) e. _V )

Syntax hints:   -> wi 4   A.wal 1371   = wceq 1373   e. wcel 2177   {cab 2192   _Vcvv 2773   C_ wss 3170   U.cuni 3859   iotacio 5244

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-un 4493

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-uni 3860  df-iota 5246

This theorem is referenced by:  fngsum  13305  igsumvalx  13306