Theorem setrec1 42438

Description: This is the first of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs ( F ) is closed under F. This effectively sets the actual value of setrecs ( F ) as a lower bound for setrecs ( F ), as it implies that any set generated by successive applications of F is a member of B. This theorem "gets off the ground" because we can start by letting A = (/), and the hypotheses of the theorem will hold trivially.

Variable B represents an abbreviation of setrecs ( F ) or another name of setrecs ( F ) (for an example of the latter, see theorem setrecon).

Proof summary: Assume that A C_ B, meaning that all elements of A are in some set recursively generated by F. Then by setrec1lem3 42436, A is a subset of some set recursively generated by F. (It turns out that A itself is recursively generated by F, but we don't need this fact. See the comment to setrec1lem3 42436.) Therefore, by setrec1lem4 42437, ( F ` A ) is a subset of some set recursively generated by F. Thus, by ssuni 4459, it is a subset of the union of all sets recursively generated by F.

See df-setrecs 42431 for a detailed description of how the setrecs definition works.

(Contributed by Emmett Weisz, 9-Oct-2020.)

Hypotheses

Ref	Expression
setrec1.b	|- B = setrecs ( F )
setrec1.v	|- ( ph -> A e. _V )
setrec1.a	|- ( ph -> A C_ B )

Assertion

Ref	Expression
setrec1	|- ( ph -> ( F ` A ) C_ B )

Proof of Theorem setrec1

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- { 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 -> ( F ` w ) C_ z ) ) -> y C_ z ) }
2		setrec1.v	. . . 4 |- ( ph -> A e. _V )
3		setrec1.a	. . . . . . . . 9 |- ( ph -> A C_ B )
4	3	sseld 3602	. . . . . . . 8 |- ( ph -> ( a e. A -> a e. B ) )
5	4	imp 445	. . . . . . 7 |- ( ( ph /\ a e. A ) -> a e. B )
6		setrec1.b	. . . . . . . 8 |- B = setrecs ( F )
7		df-setrecs 42431	. . . . . . . 8 |- setrecs ( F ) = U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
8	6, 7	eqtri 2644	. . . . . . 7 |- B = U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
9	5, 8	syl6eleq 2711	. . . . . 6 |- ( ( ph /\ a e. A ) -> a e. U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } )
10		eluni 4439	. . . . . 6 |- ( a e. U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } <-> E. x ( a e. x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) )
11	9, 10	sylib 208	. . . . 5 |- ( ( ph /\ a e. A ) -> E. x ( a e. x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) )
12	11	ralrimiva 2966	. . . 4 |- ( ph -> A. a e. A E. x ( a e. x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) )
13	1, 2, 12	setrec1lem3 42436	. . 3 |- ( ph -> E. x ( A C_ x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) )
14		nfv 1843	. . . . . . 7 |- F/ z ph
15		nfv 1843	. . . . . . . 8 |- F/ z A C_ x
16		nfaba1 2770	. . . . . . . . 9 |- F/_ z { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
17	16	nfel2 2781	. . . . . . . 8 |- F/ z x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
18	15, 17	nfan 1828	. . . . . . 7 |- F/ z ( A C_ x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } )
19	14, 18	nfan 1828	. . . . . 6 |- F/ z ( ph /\ ( A C_ x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) )
20	2	adantr 481	. . . . . 6 |- ( ( ph /\ ( A C_ x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) ) -> A e. _V )
21		simprl 794	. . . . . 6 |- ( ( ph /\ ( A C_ x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) ) -> A C_ x )
22		simprr 796	. . . . . 6 |- ( ( ph /\ ( A C_ x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) ) -> x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } )
23	19, 1, 20, 21, 22	setrec1lem4 42437	. . . . 5 |- ( ( ph /\ ( A C_ x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) ) -> ( x u. ( F ` A ) ) e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } )
24		ssun2 3777	. . . . 5 |- ( F ` A ) C_ ( x u. ( F ` A ) )
25	23, 24	jctil 560	. . . 4 |- ( ( ph /\ ( A C_ x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) ) -> ( ( F ` A ) C_ ( x u. ( F ` A ) ) /\ ( x u. ( F ` A ) ) e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) )
26		ssuni 4459	. . . 4 |- ( ( ( F ` A ) C_ ( x u. ( F ` A ) ) /\ ( x u. ( F ` A ) ) e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) -> ( F ` A ) C_ U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } )
27	25, 26	syl 17	. . 3 |- ( ( ph /\ ( A C_ x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) ) -> ( F ` A ) C_ U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } )
28	13, 27	exlimddv 1863	. 2 |- ( ph -> ( F ` A ) C_ U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } )
29	28, 8	syl6sseqr 3652	1 |- ( ph -> ( F ` A ) C_ B )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   u. cun 3572   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-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  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628  df-setrecs 42431

This theorem is referenced by:  elsetrecslem  42444  elsetrecs  42445  vsetrec  42446  onsetrec  42451