Theorem iunssd 3947

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 2563	. 2 |- ( ph -> A. x e. A B C_ C )
3		iunss 3942	. 2 |- ( U_ x e. A B C_ C <-> A. x e. A B C_ C )
4	2, 3	sylibr 134	1 |- ( ph -> U_ x e. A B C_ C )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2160   A.wral 2468   C_ wss 3144   U_ciun 3901

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-iun 3903

This theorem is referenced by:  imasaddfnlemg  12788  imasaddflemg  12790