Theorem iotaexel 5927

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 5922	. . 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 1494	. . . . 5 |- ( A. x ( ph -> x e. A ) <-> A. x ( ph <-> ( x e. A /\ ph ) ) )
4		iotabi 5260	. . . . 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 2259	. 2 |- ( ( A e. V /\ A. x ( ph -> x e. A ) ) -> ( iota_ x e. A ph ) = ( iota x ph ) )
8		riotaexg 5926	. . 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 2285	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 2178   _Vcvv 2776   iotacio 5249   iota_crio 5921

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 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-iota 5251  df-riota 5922

This theorem is referenced by: (None)