Theorem iunssd 39271

Description: Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypothesis

Ref	Expression
iunssd.1	|- ( ( ph /\ x e. A ) -> B C_ C )

Assertion

Ref	Expression
iunssd	|- ( ph -> U_ x e. A B C_ C )

Distinct variable groups:   x, C   ph, x

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem iunssd

Step	Hyp	Ref	Expression
1		iunssd.1	. . 3 |- ( ( ph /\ x e. A ) -> B C_ C )
2	1	ralrimiva 2966	. 2 |- ( ph -> A. x e. A B C_ C )
3		iunss 4561	. 2 |- ( U_ x e. A B C_ C <-> A. x e. A B C_ C )
4	2, 3	sylibr 224	1 |- ( ph -> U_ x e. A B C_ C )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   C_ wss 3574   U_ciun 4520

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-iun 4522

This theorem is referenced by:  meaiininclem  40700  smflim  40985  smfresal  40995  smfmullem4  41001