Theorem mreexdomd 16310

Description: In a Moore system whose closure operator has the exchange property, if S is independent and contained in the closure of T, and either S or T is finite, then T dominates S. This is an immediate consequence of mreexexd 16308. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
mreexdomd.1	|- ( ph -> A e. (Moore ` X ) )
mreexdomd.2	|- N = (mrCls ` A )
mreexdomd.3	|- I = (mrInd ` A )
mreexdomd.4	|- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )
mreexdomd.5	|- ( ph -> S C_ ( N ` T ) )
mreexdomd.6	|- ( ph -> T C_ X )
mreexdomd.7	|- ( ph -> ( S e. Fin \/ T e. Fin ) )
mreexdomd.8	|- ( ph -> S e. I )

Assertion

Ref	Expression
mreexdomd	|- ( ph -> S ~<_ T )

Distinct variable groups:   X, s, y, z   ph, s, y, z   I, s, y, z   N, s, y, z

Allowed substitution hints:   A( y, z, s)   S( y, z, s)   T( y, z, s)

Proof of Theorem mreexdomd

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mreexdomd.1	. . 3 |- ( ph -> A e. (Moore ` X ) )
2		mreexdomd.2	. . 3 |- N = (mrCls ` A )
3		mreexdomd.3	. . 3 |- I = (mrInd ` A )
4		mreexdomd.4	. . 3 |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )
5		mreexdomd.8	. . . . 5 |- ( ph -> S e. I )
6	3, 1, 5	mrissd 16296	. . . 4 |- ( ph -> S C_ X )
7		dif0 3950	. . . 4 |- ( X \ (/) ) = X
8	6, 7	syl6sseqr 3652	. . 3 |- ( ph -> S C_ ( X \ (/) ) )
9		mreexdomd.6	. . . 4 |- ( ph -> T C_ X )
10	9, 7	syl6sseqr 3652	. . 3 |- ( ph -> T C_ ( X \ (/) ) )
11		mreexdomd.5	. . . 4 |- ( ph -> S C_ ( N ` T ) )
12		un0 3967	. . . . 5 |- ( T u. (/) ) = T
13	12	fveq2i 6194	. . . 4 |- ( N ` ( T u. (/) ) ) = ( N ` T )
14	11, 13	syl6sseqr 3652	. . 3 |- ( ph -> S C_ ( N ` ( T u. (/) ) ) )
15		un0 3967	. . . 4 |- ( S u. (/) ) = S
16	15, 5	syl5eqel 2705	. . 3 |- ( ph -> ( S u. (/) ) e. I )
17		mreexdomd.7	. . 3 |- ( ph -> ( S e. Fin \/ T e. Fin ) )
18	1, 2, 3, 4, 8, 10, 14, 16, 17	mreexexd 16308	. 2 |- ( ph -> E. i e. ~P T ( S ~~ i /\ ( i u. (/) ) e. I ) )
19		simprrl 804	. . 3 |- ( ( ph /\ ( i e. ~P T /\ ( S ~~ i /\ ( i u. (/) ) e. I ) ) ) -> S ~~ i )
20		simprl 794	. . . . 5 |- ( ( ph /\ ( i e. ~P T /\ ( S ~~ i /\ ( i u. (/) ) e. I ) ) ) -> i e. ~P T )
21	20	elpwid 4170	. . . 4 |- ( ( ph /\ ( i e. ~P T /\ ( S ~~ i /\ ( i u. (/) ) e. I ) ) ) -> i C_ T )
22	1	elfvexd 6222	. . . . . . 7 |- ( ph -> X e. _V )
23	22, 9	ssexd 4805	. . . . . 6 |- ( ph -> T e. _V )
24		ssdomg 8001	. . . . . 6 |- ( T e. _V -> ( i C_ T -> i ~<_ T ) )
25	23, 24	syl 17	. . . . 5 |- ( ph -> ( i C_ T -> i ~<_ T ) )
26	25	adantr 481	. . . 4 |- ( ( ph /\ ( i e. ~P T /\ ( S ~~ i /\ ( i u. (/) ) e. I ) ) ) -> ( i C_ T -> i ~<_ T ) )
27	21, 26	mpd 15	. . 3 |- ( ( ph /\ ( i e. ~P T /\ ( S ~~ i /\ ( i u. (/) ) e. I ) ) ) -> i ~<_ T )
28		endomtr 8014	. . 3 |- ( ( S ~~ i /\ i ~<_ T ) -> S ~<_ T )
29	19, 27, 28	syl2anc 693	. 2 |- ( ( ph /\ ( i e. ~P T /\ ( S ~~ i /\ ( i u. (/) ) e. I ) ) ) -> S ~<_ T )
30	18, 29	rexlimddv 3035	1 |- ( ph -> S ~<_ T )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955  Moorecmre 16242  mrClscmrc 16243  mrIndcmri 16244

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-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-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-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-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-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-mre 16246  df-mrc 16247  df-mri 16248

This theorem is referenced by:  mreexfidimd  16311