Theorem bj-intabssel1 12578

Description: Version of intss1 3733 using a class abstraction and implicit substitution. Closed form of intmin3 3745. (Contributed by BJ, 29-Nov-2019.)

Hypotheses

Ref	Expression
bj-intabssel1.nf	|- F/_ x A
bj-intabssel1.nf2	|- F/ x ps
bj-intabssel1.is	|- ( x = A -> ( ps -> ph ) )

Assertion

Ref	Expression
bj-intabssel1	|- ( A e. V -> ( ps -> |^| { x | ph } C_ A ) )

Proof of Theorem bj-intabssel1

Step	Hyp	Ref	Expression
1		bj-intabssel1.nf	. . 3 |- F/_ x A
2		bj-intabssel1.nf2	. . 3 |- F/ x ps
3		bj-intabssel1.is	. . 3 |- ( x = A -> ( ps -> ph ) )
4	1, 2, 3	elabgf2 12568	. 2 |- ( A e. V -> ( ps -> A e. { x | ph } ) )
5		intss1 3733	. 2 |- ( A e. { x | ph } -> |^| { x | ph } C_ A )
6	4, 5	syl6 33	1 |- ( A e. V -> ( ps -> |^| { x | ph } C_ A ) )

Syntax hints:   -> wi 4   = wceq 1299   F/wnf 1404   e. wcel 1448   {cab 2086   F/_wnfc 2227   C_ wss 3021   |^|cint 3718

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-ss 3034  df-int 3719

This theorem is referenced by:  bj-omssind  12718