Theorem setrec1lem1 42434

Description: Lemma for setrec1 42438. This is a utility theorem showing the equivalence of the statement X e. Y and its expanded form. The proof uses elabg 3351 and equivalence theorems.

Variable Y is the class of sets y that are recursively generated by the function F. In other words, y e. Y iff by starting with the empty set and repeatedly applying F to subsets w of our set, we will eventually generate all the elements of Y. In this theorem, X is any element of Y, and V is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
setrec1lem1.1	|- Y = { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
setrec1lem1.2	|- ( ph -> X e. V )

Assertion

Ref	Expression
setrec1lem1	|- ( ph -> ( X e. Y <-> A. z ( A. w ( w C_ X -> ( w C_ z -> ( F ` w ) C_ z ) ) -> X C_ z ) ) )

Distinct variable groups:   y, F   w, X, y   z, X, y

Allowed substitution hints:   ph( y, z, w)   F( z, w)   V( y, z, w)   Y( y, z, w)

Proof of Theorem setrec1lem1

Step	Hyp	Ref	Expression
1		setrec1lem1.2	. 2 |- ( ph -> X e. V )
2		sseq2 3627	. . . . . . 7 |- ( y = X -> ( w C_ y <-> w C_ X ) )
3	2	imbi1d 331	. . . . . 6 |- ( y = X -> ( ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) <-> ( w C_ X -> ( w C_ z -> ( F ` w ) C_ z ) ) ) )
4	3	albidv 1849	. . . . 5 |- ( y = X -> ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) <-> A. w ( w C_ X -> ( w C_ z -> ( F ` w ) C_ z ) ) ) )
5		sseq1 3626	. . . . 5 |- ( y = X -> ( y C_ z <-> X C_ z ) )
6	4, 5	imbi12d 334	. . . 4 |- ( y = X -> ( ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) <-> ( A. w ( w C_ X -> ( w C_ z -> ( F ` w ) C_ z ) ) -> X C_ z ) ) )
7	6	albidv 1849	. . 3 |- ( y = X -> ( A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) <-> A. z ( A. w ( w C_ X -> ( w C_ z -> ( F ` w ) C_ z ) ) -> X C_ z ) ) )
8		setrec1lem1.1	. . 3 |- Y = { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
9	7, 8	elab2g 3353	. 2 |- ( X e. V -> ( X e. Y <-> A. z ( A. w ( w C_ X -> ( w C_ z -> ( F ` w ) C_ z ) ) -> X C_ z ) ) )
10	1, 9	syl 17	1 |- ( ph -> ( X e. Y <-> A. z ( A. w ( w C_ X -> ( w C_ z -> ( F ` w ) C_ z ) ) -> X C_ z ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   C_ wss 3574   ` cfv 5888

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-v 3202  df-in 3581  df-ss 3588

This theorem is referenced by:  setrec1lem2  42435  setrec1lem4  42437  setrec2fun  42439