Theorem mreexexlem3d 16306

Description: Base case of the induction in mreexexd 16308. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
mreexexlem2d.1	|- ( ph -> A e. (Moore ` X ) )
mreexexlem2d.2	|- N = (mrCls ` A )
mreexexlem2d.3	|- I = (mrInd ` A )
mreexexlem2d.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 } ) ) )
mreexexlem2d.5	|- ( ph -> F C_ ( X \ H ) )
mreexexlem2d.6	|- ( ph -> G C_ ( X \ H ) )
mreexexlem2d.7	|- ( ph -> F C_ ( N ` ( G u. H ) ) )
mreexexlem2d.8	|- ( ph -> ( F u. H ) e. I )
mreexexlem3d.9	|- ( ph -> ( F = (/) \/ G = (/) ) )

Assertion

Ref	Expression
mreexexlem3d	|- ( ph -> E. i e. ~P G ( F ~~ i /\ ( i u. H ) e. I ) )

Distinct variable groups:   i, F   i, G   i, H   i, I

Allowed substitution hints:   ph( y, z, i, s)   A( y, z, i, s)   F( y, z, s)   G( y, z, s)   H( y, z, s)   I( y, z, s)   N( y, z, i, s)   X( y, z, i, s)

Proof of Theorem mreexexlem3d

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( ph /\ F = (/) ) -> F = (/) )
2		mreexexlem2d.1	. . . . . . . . . 10 |- ( ph -> A e. (Moore ` X ) )
3	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ G = (/) ) -> A e. (Moore ` X ) )
4		mreexexlem2d.2	. . . . . . . . 9 |- N = (mrCls ` A )
5		mreexexlem2d.3	. . . . . . . . 9 |- I = (mrInd ` A )
6		mreexexlem2d.7	. . . . . . . . . . . 12 |- ( ph -> F C_ ( N ` ( G u. H ) ) )
7	6	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ G = (/) ) -> F C_ ( N ` ( G u. H ) ) )
8		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ph /\ G = (/) ) -> G = (/) )
9	8	uneq1d 3766	. . . . . . . . . . . . 13 |- ( ( ph /\ G = (/) ) -> ( G u. H ) = ( (/) u. H ) )
10		uncom 3757	. . . . . . . . . . . . . 14 |- ( H u. (/) ) = ( (/) u. H )
11		un0 3967	. . . . . . . . . . . . . 14 |- ( H u. (/) ) = H
12	10, 11	eqtr3i 2646	. . . . . . . . . . . . 13 |- ( (/) u. H ) = H
13	9, 12	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( ph /\ G = (/) ) -> ( G u. H ) = H )
14	13	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ph /\ G = (/) ) -> ( N ` ( G u. H ) ) = ( N ` H ) )
15	7, 14	sseqtrd 3641	. . . . . . . . . 10 |- ( ( ph /\ G = (/) ) -> F C_ ( N ` H ) )
16		mreexexlem2d.8	. . . . . . . . . . . . . 14 |- ( ph -> ( F u. H ) e. I )
17	16	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ G = (/) ) -> ( F u. H ) e. I )
18	5, 3, 17	mrissd 16296	. . . . . . . . . . . 12 |- ( ( ph /\ G = (/) ) -> ( F u. H ) C_ X )
19	18	unssbd 3791	. . . . . . . . . . 11 |- ( ( ph /\ G = (/) ) -> H C_ X )
20	3, 4, 19	mrcssidd 16285	. . . . . . . . . 10 |- ( ( ph /\ G = (/) ) -> H C_ ( N ` H ) )
21	15, 20	unssd 3789	. . . . . . . . 9 |- ( ( ph /\ G = (/) ) -> ( F u. H ) C_ ( N ` H ) )
22		ssun2 3777	. . . . . . . . . 10 |- H C_ ( F u. H )
23	22	a1i 11	. . . . . . . . 9 |- ( ( ph /\ G = (/) ) -> H C_ ( F u. H ) )
24	3, 4, 5, 21, 23, 17	mrissmrcd 16300	. . . . . . . 8 |- ( ( ph /\ G = (/) ) -> ( F u. H ) = H )
25		ssequn1 3783	. . . . . . . 8 |- ( F C_ H <-> ( F u. H ) = H )
26	24, 25	sylibr 224	. . . . . . 7 |- ( ( ph /\ G = (/) ) -> F C_ H )
27		mreexexlem2d.5	. . . . . . . 8 |- ( ph -> F C_ ( X \ H ) )
28	27	adantr 481	. . . . . . 7 |- ( ( ph /\ G = (/) ) -> F C_ ( X \ H ) )
29	26, 28	ssind 3837	. . . . . 6 |- ( ( ph /\ G = (/) ) -> F C_ ( H i^i ( X \ H ) ) )
30		disjdif 4040	. . . . . 6 |- ( H i^i ( X \ H ) ) = (/)
31	29, 30	syl6sseq 3651	. . . . 5 |- ( ( ph /\ G = (/) ) -> F C_ (/) )
32		ss0b 3973	. . . . 5 |- ( F C_ (/) <-> F = (/) )
33	31, 32	sylib 208	. . . 4 |- ( ( ph /\ G = (/) ) -> F = (/) )
34		mreexexlem3d.9	. . . 4 |- ( ph -> ( F = (/) \/ G = (/) ) )
35	1, 33, 34	mpjaodan 827	. . 3 |- ( ph -> F = (/) )
36		0elpw 4834	. . 3 |- (/) e. ~P G
37	35, 36	syl6eqel 2709	. 2 |- ( ph -> F e. ~P G )
38	2	elfvexd 6222	. . . 4 |- ( ph -> X e. _V )
39	27	difss2d 3740	. . . 4 |- ( ph -> F C_ X )
40	38, 39	ssexd 4805	. . 3 |- ( ph -> F e. _V )
41		enrefg 7987	. . 3 |- ( F e. _V -> F ~~ F )
42	40, 41	syl 17	. 2 |- ( ph -> F ~~ F )
43		breq2 4657	. . . 4 |- ( i = F -> ( F ~~ i <-> F ~~ F ) )
44		uneq1 3760	. . . . 5 |- ( i = F -> ( i u. H ) = ( F u. H ) )
45	44	eleq1d 2686	. . . 4 |- ( i = F -> ( ( i u. H ) e. I <-> ( F u. H ) e. I ) )
46	43, 45	anbi12d 747	. . 3 |- ( i = F -> ( ( F ~~ i /\ ( i u. H ) e. I ) <-> ( F ~~ F /\ ( F u. H ) e. I ) ) )
47	46	rspcev 3309	. 2 |- ( ( F e. ~P G /\ ( F ~~ F /\ ( F u. H ) e. I ) ) -> E. i e. ~P G ( F ~~ i /\ ( i u. H ) e. I ) )
48	37, 42, 16, 47	syl12anc 1324	1 |- ( ph -> E. i e. ~P G ( F ~~ i /\ ( i u. H ) e. I ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   ` cfv 5888   ~~ cen 7952  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-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-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-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-en 7956  df-mre 16246  df-mrc 16247  df-mri 16248

This theorem is referenced by:  mreexexlem4d  16307  mreexexd  16308  mreexexdOLD  16309