Theorem frege109d 38049

Description: If A contains all elements of U and all elements after those in U in the transitive closure of R, then the image under R of A is a subclass of A. Similar to Proposition 109 of [Frege1879] p. 74. Compare with frege109 38266. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege109d.r	|- ( ph -> R e. _V )
frege109d.a	|- ( ph -> A = ( U u. ( ( t+ ` R ) " U ) ) )

Assertion

Ref	Expression
frege109d	|- ( ph -> ( R " A ) C_ A )

Proof of Theorem frege109d

Step	Hyp	Ref	Expression
1		frege109d.r	. . . . 5 |- ( ph -> R e. _V )
2		trclfvlb 13749	. . . . 5 |- ( R e. _V -> R C_ ( t+ ` R ) )
3		imass1 5500	. . . . 5 |- ( R C_ ( t+ ` R ) -> ( R " U ) C_ ( ( t+ ` R ) " U ) )
4	1, 2, 3	3syl 18	. . . 4 |- ( ph -> ( R " U ) C_ ( ( t+ ` R ) " U ) )
5		coss1 5277	. . . . . . 7 |- ( R C_ ( t+ ` R ) -> ( R o. ( t+ ` R ) ) C_ ( ( t+ ` R ) o. ( t+ ` R ) ) )
6	1, 2, 5	3syl 18	. . . . . 6 |- ( ph -> ( R o. ( t+ ` R ) ) C_ ( ( t+ ` R ) o. ( t+ ` R ) ) )
7		trclfvcotrg 13757	. . . . . 6 |- ( ( t+ ` R ) o. ( t+ ` R ) ) C_ ( t+ ` R )
8	6, 7	syl6ss 3615	. . . . 5 |- ( ph -> ( R o. ( t+ ` R ) ) C_ ( t+ ` R ) )
9		imass1 5500	. . . . 5 |- ( ( R o. ( t+ ` R ) ) C_ ( t+ ` R ) -> ( ( R o. ( t+ ` R ) ) " U ) C_ ( ( t+ ` R ) " U ) )
10	8, 9	syl 17	. . . 4 |- ( ph -> ( ( R o. ( t+ ` R ) ) " U ) C_ ( ( t+ ` R ) " U ) )
11	4, 10	unssd 3789	. . 3 |- ( ph -> ( ( R " U ) u. ( ( R o. ( t+ ` R ) ) " U ) ) C_ ( ( t+ ` R ) " U ) )
12		ssun2 3777	. . 3 |- ( ( t+ ` R ) " U ) C_ ( U u. ( ( t+ ` R ) " U ) )
13	11, 12	syl6ss 3615	. 2 |- ( ph -> ( ( R " U ) u. ( ( R o. ( t+ ` R ) ) " U ) ) C_ ( U u. ( ( t+ ` R ) " U ) ) )
14		frege109d.a	. . . 4 |- ( ph -> A = ( U u. ( ( t+ ` R ) " U ) ) )
15	14	imaeq2d 5466	. . 3 |- ( ph -> ( R " A ) = ( R " ( U u. ( ( t+ ` R ) " U ) ) ) )
16		imaundi 5545	. . . 4 |- ( R " ( U u. ( ( t+ ` R ) " U ) ) ) = ( ( R " U ) u. ( R " ( ( t+ ` R ) " U ) ) )
17		imaco 5640	. . . . . 6 |- ( ( R o. ( t+ ` R ) ) " U ) = ( R " ( ( t+ ` R ) " U ) )
18	17	eqcomi 2631	. . . . 5 |- ( R " ( ( t+ ` R ) " U ) ) = ( ( R o. ( t+ ` R ) ) " U )
19	18	uneq2i 3764	. . . 4 |- ( ( R " U ) u. ( R " ( ( t+ ` R ) " U ) ) ) = ( ( R " U ) u. ( ( R o. ( t+ ` R ) ) " U ) )
20	16, 19	eqtri 2644	. . 3 |- ( R " ( U u. ( ( t+ ` R ) " U ) ) ) = ( ( R " U ) u. ( ( R o. ( t+ ` R ) ) " U ) )
21	15, 20	syl6eq 2672	. 2 |- ( ph -> ( R " A ) = ( ( R " U ) u. ( ( R o. ( t+ ` R ) ) " U ) ) )
22	13, 21, 14	3sstr4d 3648	1 |- ( ph -> ( R " A ) C_ A )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   "cima 5117   o. ccom 5118   ` cfv 5888   t+ctcl 13724

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-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-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  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-trcl 13726

This theorem is referenced by: (None)