Theorem iunssd 3962

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 2570	. 2 |- ( ph -> A. x e. A B C_ C )
3		iunss 3957	. 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 2167   A.wral 2475   C_ wss 3157   U_ciun 3916

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-ext 2178

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-in 3163  df-ss 3170  df-iun 3918

This theorem is referenced by:  imasaddfnlemg  12957  imasaddflemg  12959