Theorem iotaexel 5904

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 5899	. . 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 1493	. . . . 5 |- ( A. x ( ph -> x e. A ) <-> A. x ( ph <-> ( x e. A /\ ph ) ) )
4		iotabi 5241	. . . . 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 2257	. 2 |- ( ( A e. V /\ A. x ( ph -> x e. A ) ) -> ( iota_ x e. A ph ) = ( iota x ph ) )
8		riotaexg 5903	. . 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 2283	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 1371   = wceq 1373   e. wcel 2176   _Vcvv 2772   iotacio 5230   iota_crio 5898

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-iota 5232  df-riota 5899

This theorem is referenced by: (None)