Theorem iunssd 4016

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 2605	. 2 |- ( ph -> A. x e. A B C_ C )
3		iunss 4011	. 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 2202   A.wral 2510   C_ wss 3200   U_ciun 3970

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-iun 3972

This theorem is referenced by:  imasaddfnlemg  13396  imasaddflemg  13398