Theorem iinexd 39318

Description: The existence of an indexed union. x is normally a free-variable parameter in B, which should be read B ( x ). (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
iinexd.1	|- ( ph -> A =/= (/) )
iinexd.2	|- ( ph -> A. x e. A B e. C )

Assertion

Ref	Expression
iinexd	|- ( ph -> |^|_ x e. A B e. _V )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)   C( x)

Proof of Theorem iinexd

Step	Hyp	Ref	Expression
1		iinexd.1	. 2 |- ( ph -> A =/= (/) )
2		iinexd.2	. 2 |- ( ph -> A. x e. A B e. C )
3		iinexg 4824	. 2 |- ( ( A =/= (/) /\ A. x e. A B e. C ) -> |^|_ x e. A B e. _V )
4	1, 2, 3	syl2anc 693	1 |- ( ph -> |^|_ x e. A B e. _V )

Syntax hints:   -> wi 4   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   (/)c0 3915   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-int 4476  df-iin 4523

This theorem is referenced by:  smfsuplem1  41017  smfinflem  41023  smflimsuplem1  41026  smflimsuplem2  41027  smflimsuplem3  41028  smflimsuplem4  41029  smflimsuplem5  41030  smflimsuplem7  41032