Theorem setrec2 42442

Description: This is the second of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs ( F ) is a subclass of all classes C that are closed under F. Taken together, theorems setrec1 42438 and setrec2v 42443 uniquely determine setrecs ( F ) to be the minimal class closed under F.

We express this by saying that if F respects the C_ relation and C is closed under F, then B C_ C. By substituting strategically constructed classes for C, we can easily prove many useful properties. Although this theorem cannot show equality between B and C, if we intend to prove equality between B and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7053) to the other class.

(Contributed by Emmett Weisz, 2-Sep-2021.)

Hypotheses

Ref	Expression
setrec2.1	|- F/_ a F
setrec2.2	|- B = setrecs ( F )
setrec2.3	|- ( ph -> A. a ( a C_ C -> ( F ` a ) C_ C ) )

Assertion

Ref	Expression
setrec2	|- ( ph -> B C_ C )

Distinct variable group:   C, a

Allowed substitution hints:   ph( a)   B( a)   F( a)

Proof of Theorem setrec2

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

Step	Hyp	Ref	Expression
1		setrec2.1	. . 3 |- F/_ a F
2		nfcv 2764	. . . . . 6 |- F/_ a x
3		nfcv 2764	. . . . . 6 |- F/_ a u
4	2, 1, 3	nfbr 4699	. . . . 5 |- F/ a x F u
5	4	nfeu 2486	. . . 4 |- F/ a E! u x F u
6	5	nfab 2769	. . 3 |- F/_ a { x | E! u x F u }
7	1, 6	nfres 5398	. 2 |- F/_ a ( F |` { x | E! u x F u } )
8		setrec2.2	. . 3 |- B = setrecs ( F )
9		setrec2lem1 42440	. . . . . . . . . . . 12 |- ( ( F |` { x | E! u x F u } ) ` w ) = ( F ` w )
10	9	sseq1i 3629	. . . . . . . . . . 11 |- ( ( ( F |` { x | E! u x F u } ) ` w ) C_ z <-> ( F ` w ) C_ z )
11	10	imbi2i 326	. . . . . . . . . 10 |- ( ( w C_ z -> ( ( F |` { x | E! u x F u } ) ` w ) C_ z ) <-> ( w C_ z -> ( F ` w ) C_ z ) )
12	11	imbi2i 326	. . . . . . . . 9 |- ( ( w C_ y -> ( w C_ z -> ( ( F |` { x | E! u x F u } ) ` w ) C_ z ) ) <-> ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) )
13	12	albii 1747	. . . . . . . 8 |- ( A. w ( w C_ y -> ( w C_ z -> ( ( F |` { x | E! u x F u } ) ` w ) C_ z ) ) <-> A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) )
14	13	imbi1i 339	. . . . . . 7 |- ( ( A. w ( w C_ y -> ( w C_ z -> ( ( F |` { x | E! u x F u } ) ` w ) C_ z ) ) -> y C_ z ) <-> ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) )
15	14	albii 1747	. . . . . 6 |- ( A. z ( A. w ( w C_ y -> ( w C_ z -> ( ( F |` { x | E! u x F u } ) ` w ) C_ z ) ) -> y C_ z ) <-> A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) )
16	15	abbii 2739	. . . . 5 |- { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( ( F |` { x | E! u x F u } ) ` w ) C_ z ) ) -> y C_ z ) } = { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
17	16	unieqi 4445	. . . 4 |- U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( ( F |` { x | E! u x F u } ) ` w ) C_ z ) ) -> y C_ z ) } = U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
18		df-setrecs 42431	. . . 4 |- setrecs ( ( F |` { x | E! u x F u } ) ) = U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( ( F |` { x | E! u x F u } ) ` w ) C_ z ) ) -> y C_ z ) }
19		df-setrecs 42431	. . . 4 |- setrecs ( F ) = U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
20	17, 18, 19	3eqtr4ri 2655	. . 3 |- setrecs ( F ) = setrecs ( ( F |` { x | E! u x F u } ) )
21	8, 20	eqtri 2644	. 2 |- B = setrecs ( ( F |` { x | E! u x F u } ) )
22		setrec2lem2 42441	. 2 |- Fun ( F |` { x | E! u x F u } )
23		setrec2.3	. . 3 |- ( ph -> A. a ( a C_ C -> ( F ` a ) C_ C ) )
24		setrec2lem1 42440	. . . . . 6 |- ( ( F |` { x | E! u x F u } ) ` a ) = ( F ` a )
25	24	sseq1i 3629	. . . . 5 |- ( ( ( F |` { x | E! u x F u } ) ` a ) C_ C <-> ( F ` a ) C_ C )
26	25	imbi2i 326	. . . 4 |- ( ( a C_ C -> ( ( F |` { x | E! u x F u } ) ` a ) C_ C ) <-> ( a C_ C -> ( F ` a ) C_ C ) )
27	26	albii 1747	. . 3 |- ( A. a ( a C_ C -> ( ( F |` { x | E! u x F u } ) ` a ) C_ C ) <-> A. a ( a C_ C -> ( F ` a ) C_ C ) )
28	23, 27	sylibr 224	. 2 |- ( ph -> A. a ( a C_ C -> ( ( F |` { x | E! u x F u } ) ` a ) C_ C ) )
29	7, 21, 22, 28	setrec2fun 42439	1 |- ( ph -> B C_ C )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   E!weu 2470   {cab 2608   F/_wnfc 2751   C_ wss 3574   U.cuni 4436   class class class wbr 4653   |` cres 5116   ` 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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-setrecs 42431

This theorem is referenced by:  setrec2v  42443