Theorem setrecseq 42432

Description: Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.)

Assertion

Ref	Expression
setrecseq	|- ( F = G -> setrecs ( F ) = setrecs ( G ) )

Proof of Theorem setrecseq

Dummy variables y w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6190	. . . . . . . . . 10 |- ( F = G -> ( F ` w ) = ( G ` w ) )
2	1	sseq1d 3632	. . . . . . . . 9 |- ( F = G -> ( ( F ` w ) C_ z <-> ( G ` w ) C_ z ) )
3	2	imbi2d 330	. . . . . . . 8 |- ( F = G -> ( ( w C_ z -> ( F ` w ) C_ z ) <-> ( w C_ z -> ( G ` w ) C_ z ) ) )
4	3	imbi2d 330	. . . . . . 7 |- ( F = G -> ( ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) <-> ( w C_ y -> ( w C_ z -> ( G ` w ) C_ z ) ) ) )
5	4	albidv 1849	. . . . . 6 |- ( F = G -> ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) <-> A. w ( w C_ y -> ( w C_ z -> ( G ` w ) C_ z ) ) ) )
6	5	imbi1d 331	. . . . 5 |- ( F = G -> ( ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) <-> ( A. w ( w C_ y -> ( w C_ z -> ( G ` w ) C_ z ) ) -> y C_ z ) ) )
7	6	albidv 1849	. . . 4 |- ( F = G -> ( A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) <-> A. z ( A. w ( w C_ y -> ( w C_ z -> ( G ` w ) C_ z ) ) -> y C_ z ) ) )
8	7	abbidv 2741	. . 3 |- ( F = G -> { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } = { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( G ` w ) C_ z ) ) -> y C_ z ) } )
9	8	unieqd 4446	. 2 |- ( F = G -> U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } = U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( G ` w ) C_ z ) ) -> y C_ z ) } )
10		df-setrecs 42431	. 2 |- setrecs ( F ) = U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
11		df-setrecs 42431	. 2 |- setrecs ( G ) = U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( G ` w ) C_ z ) ) -> y C_ z ) }
12	9, 10, 11	3eqtr4g 2681	1 |- ( F = G -> setrecs ( F ) = setrecs ( G ) )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   {cab 2608   C_ wss 3574   U.cuni 4436   ` cfv 5888  setrecscsetrecs 42430

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-rex 2918  df-in 3581  df-ss 3588  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-setrecs 42431

This theorem is referenced by: (None)