Theorem hsmexlem6 9253

Description: Lemma for hsmex 9254. (Contributed by Stefan O'Rear, 14-Feb-2015.)

Hypotheses

Ref	Expression
hsmexlem4.x	|- X e. _V
hsmexlem4.h	|- H = ( rec ( ( z e. _V |-> (har ` ~P ( X X. z ) ) ) , (har ` ~P X ) ) |` om )
hsmexlem4.u	|- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )
hsmexlem4.s	|- S = { a e. U. ( R1 " On ) | A. b e. ( TC ` { a } ) b ~<_ X }
hsmexlem4.o	|- O = OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) )

Assertion

Ref	Expression
hsmexlem6	|- S e. _V

Distinct variable groups:   a, c, d, H   S, c, d   U, c, d   a, b, z, X   x, a, y   b, c, d, x, y, z

Allowed substitution hints:   S( x, y, z, a, b)   U( x, y, z, a, b)   H( x, y, z, b)   O( x, y, z, a, b, c, d)   X( x, y, c, d)

Proof of Theorem hsmexlem6

Step	Hyp	Ref	Expression
1		fvex 6201	. 2 |- ( R1 ` (har ` ~P ( om X. U. ran H ) ) ) e. _V
2		hsmexlem4.x	. . . . 5 |- X e. _V
3		hsmexlem4.h	. . . . 5 |- H = ( rec ( ( z e. _V |-> (har ` ~P ( X X. z ) ) ) , (har ` ~P X ) ) |` om )
4		hsmexlem4.u	. . . . 5 |- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )
5		hsmexlem4.s	. . . . 5 |- S = { a e. U. ( R1 " On ) | A. b e. ( TC ` { a } ) b ~<_ X }
6		hsmexlem4.o	. . . . 5 |- O = OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) )
7	2, 3, 4, 5, 6	hsmexlem5 9252	. . . 4 |- ( d e. S -> ( rank ` d ) e. (har ` ~P ( om X. U. ran H ) ) )
8		ssrab2 3687	. . . . . . 7 |- { a e. U. ( R1 " On ) | A. b e. ( TC ` { a } ) b ~<_ X } C_ U. ( R1 " On )
9	5, 8	eqsstri 3635	. . . . . 6 |- S C_ U. ( R1 " On )
10	9	sseli 3599	. . . . 5 |- ( d e. S -> d e. U. ( R1 " On ) )
11		harcl 8466	. . . . . 6 |- (har ` ~P ( om X. U. ran H ) ) e. On
12		r1fnon 8630	. . . . . . 7 |- R1 Fn On
13		fndm 5990	. . . . . . 7 |- ( R1 Fn On -> dom R1 = On )
14	12, 13	ax-mp 5	. . . . . 6 |- dom R1 = On
15	11, 14	eleqtrri 2700	. . . . 5 |- (har ` ~P ( om X. U. ran H ) ) e. dom R1
16		rankr1ag 8665	. . . . 5 |- ( ( d e. U. ( R1 " On ) /\ (har ` ~P ( om X. U. ran H ) ) e. dom R1 ) -> ( d e. ( R1 ` (har ` ~P ( om X. U. ran H ) ) ) <-> ( rank ` d ) e. (har ` ~P ( om X. U. ran H ) ) ) )
17	10, 15, 16	sylancl 694	. . . 4 |- ( d e. S -> ( d e. ( R1 ` (har ` ~P ( om X. U. ran H ) ) ) <-> ( rank ` d ) e. (har ` ~P ( om X. U. ran H ) ) ) )
18	7, 17	mpbird 247	. . 3 |- ( d e. S -> d e. ( R1 ` (har ` ~P ( om X. U. ran H ) ) ) )
19	18	ssriv 3607	. 2 |- S C_ ( R1 ` (har ` ~P ( om X. U. ran H ) ) )
20	1, 19	ssexi 4803	1 |- S e. _V

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   _E cep 5028   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Oncon0 5723   Fn wfn 5883   ` cfv 5888   omcom 7065   reccrdg 7505   ~<_ cdom 7953  OrdIsocoi 8414  harchar 8461   TCctc 8612   R1cr1 8625   rankcrnk 8626

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-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-rmo 2920  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-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-se 5074  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-isom 5897  df-riota 6611  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-smo 7443  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-oi 8415  df-har 8463  df-wdom 8464  df-tc 8613  df-r1 8627  df-rank 8628

This theorem is referenced by:  hsmex  9254