Theorem iotaexel 5885

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 5880	. . 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 1484	. . . . 5 |- ( A. x ( ph -> x e. A ) <-> A. x ( ph <-> ( x e. A /\ ph ) ) )
4		iotabi 5229	. . . . 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 2248	. 2 |- ( ( A e. V /\ A. x ( ph -> x e. A ) ) -> ( iota_ x e. A ph ) = ( iota x ph ) )
8		riotaexg 5884	. . 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 2274	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 1362   = wceq 1364   e. wcel 2167   _Vcvv 2763   iotacio 5218   iota_crio 5879

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 4152  ax-un 4469

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 3608  df-sn 3629  df-pr 3630  df-uni 3841  df-iota 5220  df-riota 5880

This theorem is referenced by: (None)