Theorem iinssd 39314

Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
iinssd.1	|- ( ph -> X e. A )
iinssd.2	|- ( x = X -> B = D )
iinssd.3	|- ( ph -> D C_ C )

Assertion

Ref	Expression
iinssd	|- ( ph -> |^|_ x e. A B C_ C )

Distinct variable groups:   x, A   x, C   x, D   x, X

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

Proof of Theorem iinssd

Step	Hyp	Ref	Expression
1		iinssd.1	. . 3 |- ( ph -> X e. A )
2		iinssd.3	. . 3 |- ( ph -> D C_ C )
3		iinssd.2	. . . . 5 |- ( x = X -> B = D )
4	3	sseq1d 3632	. . . 4 |- ( x = X -> ( B C_ C <-> D C_ C ) )
5	4	rspcev 3309	. . 3 |- ( ( X e. A /\ D C_ C ) -> E. x e. A B C_ C )
6	1, 2, 5	syl2anc 693	. 2 |- ( ph -> E. x e. A B C_ C )
7		iinss 4571	. 2 |- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C )
8	6, 7	syl 17	1 |- ( ph -> |^|_ x e. A B C_ C )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   |^|_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-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-iin 4523

This theorem is referenced by:  smfsuplem3  41019  smflimsuplem1  41026