Theorem iotaexel 5986

Description: Set existence of an iota expression in which all values are contained within a set. (Contributed by Jim Kingdon, 28-Jun-2025.)

Assertion

Ref	Expression
iotaexel	|- ( ( A e. V /\ A. x ( ph -> x e. A ) ) -> ( iota x ph ) e. _V )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem iotaexel

Step	Hyp	Ref	Expression
1		df-riota 5981	. . 3 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
2		pm4.71r 390	. . . . . 6 |- ( ( ph -> x e. A ) <-> ( ph <-> ( x e. A /\ ph ) ) )
3	2	albii 1519	. . . . 5 |- ( A. x ( ph -> x e. A ) <-> A. x ( ph <-> ( x e. A /\ ph ) ) )
4		iotabi 5303	. . . . 5 |- ( A. x ( ph <-> ( x e. A /\ ph ) ) -> ( iota x ph ) = ( iota x ( x e. A /\ ph ) ) )
5	3, 4	sylbi 121	. . . 4 |- ( A. x ( ph -> x e. A ) -> ( iota x ph ) = ( iota x ( x e. A /\ ph ) ) )
6	5	adantl 277	. . 3 |- ( ( A e. V /\ A. x ( ph -> x e. A ) ) -> ( iota x ph ) = ( iota x ( x e. A /\ ph ) ) )
7	1, 6	eqtr4id 2283	. 2 |- ( ( A e. V /\ A. x ( ph -> x e. A ) ) -> ( iota_ x e. A ph ) = ( iota x ph ) )
8		riotaexg 5985	. . 3 |- ( A e. V -> ( iota_ x e. A ph ) e. _V )
9	8	adantr 276	. 2 |- ( ( A e. V /\ A. x ( ph -> x e. A ) ) -> ( iota_ x e. A ph ) e. _V )
10	7, 9	eqeltrrd 2309	1 |- ( ( A e. V /\ A. x ( ph -> x e. A ) ) -> ( iota x ph ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1396   = wceq 1398   e. wcel 2202   _Vcvv 2803   iotacio 5291   iota_crio 5980

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-iota 5293  df-riota 5981

This theorem is referenced by: (None)