Theorem hsmexlem5 9252

Description: Lemma for hsmex 9254. Combining the above constraints, along with itunitc 9243 and tcrank 8747, gives an effective constraint on the rank of S. (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
hsmexlem5	|- ( d e. S -> ( rank ` d ) e. (har ` ~P ( om X. U. ran H ) ) )

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 hsmexlem5

Step	Hyp	Ref	Expression
1		hsmexlem4.s	. . . . . . . 8 |- S = { a e. U. ( R1 " On ) | A. b e. ( TC ` { a } ) b ~<_ X }
2		ssrab2 3687	. . . . . . . 8 |- { a e. U. ( R1 " On ) | A. b e. ( TC ` { a } ) b ~<_ X } C_ U. ( R1 " On )
3	1, 2	eqsstri 3635	. . . . . . 7 |- S C_ U. ( R1 " On )
4	3	sseli 3599	. . . . . 6 |- ( d e. S -> d e. U. ( R1 " On ) )
5		tcrank 8747	. . . . . 6 |- ( d e. U. ( R1 " On ) -> ( rank ` d ) = ( rank " ( TC ` d ) ) )
6	4, 5	syl 17	. . . . 5 |- ( d e. S -> ( rank ` d ) = ( rank " ( TC ` d ) ) )
7		hsmexlem4.u	. . . . . . . . 9 |- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )
8	7	itunifn 9239	. . . . . . . 8 |- ( d e. S -> ( U ` d ) Fn om )
9		fniunfv 6505	. . . . . . . 8 |- ( ( U ` d ) Fn om -> U_ c e. om ( ( U ` d ) ` c ) = U. ran ( U ` d ) )
10	8, 9	syl 17	. . . . . . 7 |- ( d e. S -> U_ c e. om ( ( U ` d ) ` c ) = U. ran ( U ` d ) )
11	7	itunitc 9243	. . . . . . 7 |- ( TC ` d ) = U. ran ( U ` d )
12	10, 11	syl6reqr 2675	. . . . . 6 |- ( d e. S -> ( TC ` d ) = U_ c e. om ( ( U ` d ) ` c ) )
13	12	imaeq2d 5466	. . . . 5 |- ( d e. S -> ( rank " ( TC ` d ) ) = ( rank " U_ c e. om ( ( U ` d ) ` c ) ) )
14		imaiun 6503	. . . . . 6 |- ( rank " U_ c e. om ( ( U ` d ) ` c ) ) = U_ c e. om ( rank " ( ( U ` d ) ` c ) )
15	14	a1i 11	. . . . 5 |- ( d e. S -> ( rank " U_ c e. om ( ( U ` d ) ` c ) ) = U_ c e. om ( rank " ( ( U ` d ) ` c ) ) )
16	6, 13, 15	3eqtrd 2660	. . . 4 |- ( d e. S -> ( rank ` d ) = U_ c e. om ( rank " ( ( U ` d ) ` c ) ) )
17		dmresi 5457	. . . 4 |- dom ( _I |` U_ c e. om ( rank " ( ( U ` d ) ` c ) ) ) = U_ c e. om ( rank " ( ( U ` d ) ` c ) )
18	16, 17	syl6eqr 2674	. . 3 |- ( d e. S -> ( rank ` d ) = dom ( _I |` U_ c e. om ( rank " ( ( U ` d ) ` c ) ) ) )
19		rankon 8658	. . . . . 6 |- ( rank ` d ) e. On
20	16, 19	syl6eqelr 2710	. . . . 5 |- ( d e. S -> U_ c e. om ( rank " ( ( U ` d ) ` c ) ) e. On )
21		eloni 5733	. . . . 5 |- ( U_ c e. om ( rank " ( ( U ` d ) ` c ) ) e. On -> Ord U_ c e. om ( rank " ( ( U ` d ) ` c ) ) )
22		oiid 8446	. . . . 5 |- ( Ord U_ c e. om ( rank " ( ( U ` d ) ` c ) ) -> OrdIso ( _E , U_ c e. om ( rank " ( ( U ` d ) ` c ) ) ) = ( _I |` U_ c e. om ( rank " ( ( U ` d ) ` c ) ) ) )
23	20, 21, 22	3syl 18	. . . 4 |- ( d e. S -> OrdIso ( _E , U_ c e. om ( rank " ( ( U ` d ) ` c ) ) ) = ( _I |` U_ c e. om ( rank " ( ( U ` d ) ` c ) ) ) )
24	23	dmeqd 5326	. . 3 |- ( d e. S -> dom OrdIso ( _E , U_ c e. om ( rank " ( ( U ` d ) ` c ) ) ) = dom ( _I |` U_ c e. om ( rank " ( ( U ` d ) ` c ) ) ) )
25	18, 24	eqtr4d 2659	. 2 |- ( d e. S -> ( rank ` d ) = dom OrdIso ( _E , U_ c e. om ( rank " ( ( U ` d ) ` c ) ) ) )
26		omex 8540	. . . 4 |- om e. _V
27		wdomref 8477	. . . 4 |- ( om e. _V -> om ~<_* om )
28	26, 27	mp1i 13	. . 3 |- ( d e. S -> om ~<_* om )
29		frfnom 7530	. . . . . . 7 |- ( rec ( ( z e. _V |-> (har ` ~P ( X X. z ) ) ) , (har ` ~P X ) ) |` om ) Fn om
30		hsmexlem4.h	. . . . . . . 8 |- H = ( rec ( ( z e. _V |-> (har ` ~P ( X X. z ) ) ) , (har ` ~P X ) ) |` om )
31	30	fneq1i 5985	. . . . . . 7 |- ( H Fn om <-> ( rec ( ( z e. _V |-> (har ` ~P ( X X. z ) ) ) , (har ` ~P X ) ) |` om ) Fn om )
32	29, 31	mpbir 221	. . . . . 6 |- H Fn om
33		fniunfv 6505	. . . . . 6 |- ( H Fn om -> U_ a e. om ( H ` a ) = U. ran H )
34	32, 33	ax-mp 5	. . . . 5 |- U_ a e. om ( H ` a ) = U. ran H
35		iunon 7436	. . . . . . 7 |- ( ( om e. _V /\ A. a e. om ( H ` a ) e. On ) -> U_ a e. om ( H ` a ) e. On )
36	26, 35	mpan 706	. . . . . 6 |- ( A. a e. om ( H ` a ) e. On -> U_ a e. om ( H ` a ) e. On )
37	30	hsmexlem9 9247	. . . . . 6 |- ( a e. om -> ( H ` a ) e. On )
38	36, 37	mprg 2926	. . . . 5 |- U_ a e. om ( H ` a ) e. On
39	34, 38	eqeltrri 2698	. . . 4 |- U. ran H e. On
40	39	a1i 11	. . 3 |- ( d e. S -> U. ran H e. On )
41		fvssunirn 6217	. . . . . 6 |- ( H ` c ) C_ U. ran H
42		hsmexlem4.x	. . . . . . . 8 |- X e. _V
43		eqid 2622	. . . . . . . 8 |- OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) ) = OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) )
44	42, 30, 7, 1, 43	hsmexlem4 9251	. . . . . . 7 |- ( ( c e. om /\ d e. S ) -> dom OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) ) e. ( H ` c ) )
45	44	ancoms 469	. . . . . 6 |- ( ( d e. S /\ c e. om ) -> dom OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) ) e. ( H ` c ) )
46	41, 45	sseldi 3601	. . . . 5 |- ( ( d e. S /\ c e. om ) -> dom OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) ) e. U. ran H )
47		imassrn 5477	. . . . . . 7 |- ( rank " ( ( U ` d ) ` c ) ) C_ ran rank
48		rankf 8657	. . . . . . . 8 |- rank : U. ( R1 " On ) --> On
49		frn 6053	. . . . . . . 8 |- ( rank : U. ( R1 " On ) --> On -> ran rank C_ On )
50	48, 49	ax-mp 5	. . . . . . 7 |- ran rank C_ On
51	47, 50	sstri 3612	. . . . . 6 |- ( rank " ( ( U ` d ) ` c ) ) C_ On
52		ffun 6048	. . . . . . . 8 |- ( rank : U. ( R1 " On ) --> On -> Fun rank )
53		fvex 6201	. . . . . . . . 9 |- ( ( U ` d ) ` c ) e. _V
54	53	funimaex 5976	. . . . . . . 8 |- ( Fun rank -> ( rank " ( ( U ` d ) ` c ) ) e. _V )
55	48, 52, 54	mp2b 10	. . . . . . 7 |- ( rank " ( ( U ` d ) ` c ) ) e. _V
56	55	elpw 4164	. . . . . 6 |- ( ( rank " ( ( U ` d ) ` c ) ) e. ~P On <-> ( rank " ( ( U ` d ) ` c ) ) C_ On )
57	51, 56	mpbir 221	. . . . 5 |- ( rank " ( ( U ` d ) ` c ) ) e. ~P On
58	46, 57	jctil 560	. . . 4 |- ( ( d e. S /\ c e. om ) -> ( ( rank " ( ( U ` d ) ` c ) ) e. ~P On /\ dom OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) ) e. U. ran H ) )
59	58	ralrimiva 2966	. . 3 |- ( d e. S -> A. c e. om ( ( rank " ( ( U ` d ) ` c ) ) e. ~P On /\ dom OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) ) e. U. ran H ) )
60		eqid 2622	. . . 4 |- OrdIso ( _E , U_ c e. om ( rank " ( ( U ` d ) ` c ) ) ) = OrdIso ( _E , U_ c e. om ( rank " ( ( U ` d ) ` c ) ) )
61	43, 60	hsmexlem3 9250	. . 3 |- ( ( ( om ~<_* om /\ U. ran H e. On ) /\ A. c e. om ( ( rank " ( ( U ` d ) ` c ) ) e. ~P On /\ dom OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) ) e. U. ran H ) ) -> dom OrdIso ( _E , U_ c e. om ( rank " ( ( U ` d ) ` c ) ) ) e. (har ` ~P ( om X. U. ran H ) ) )
62	28, 40, 59, 61	syl21anc 1325	. 2 |- ( d e. S -> dom OrdIso ( _E , U_ c e. om ( rank " ( ( U ` d ) ` c ) ) ) e. (har ` ~P ( om X. U. ran H ) ) )
63	25, 62	eqeltrd 2701	1 |- ( d e. S -> ( rank ` d ) e. (har ` ~P ( om X. U. ran H ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   _E cep 5028   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Ord word 5722   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   omcom 7065   reccrdg 7505   ~<_ cdom 7953  OrdIsocoi 8414  harchar 8461   ~<_^* cwdom 8462   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:  hsmexlem6  9253