Metamath Proof Explorer

Theorem hsmexlem5

Description: Lemma for hsmex . Combining the above constraints, along with itunitc and tcrank , gives an effective constraint on the rank of S . (Contributed by Stefan O'Rear, 14-Feb-2015)

Ref Expression
Hypotheses 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 hsmexlem5 
|- ( d e. S -> ( rank ` d ) e. ( har ` ~P ( _om X. U. ran H ) ) )

Proof

Step Hyp Ref Expression
1 hsmexlem4.x 
 |-  X e. _V
2 hsmexlem4.h 
 |-  H = ( rec ( ( z e. _V |-> ( har ` ~P ( X X. z ) ) ) , ( har ` ~P X ) ) |` _om )
3 hsmexlem4.u 
 |-  U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` _om ) )
4 hsmexlem4.s 
 |-  S = { a e. U. ( R1 " On ) | A. b e. ( TC ` { a } ) b ~<_ X }
5 hsmexlem4.o 
 |-  O = OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) )
6 4 ssrab3 
 |-  S C_ U. ( R1 " On )
7 6 sseli 
 |-  ( d e. S -> d e. U. ( R1 " On ) )
8 tcrank 
 |-  ( d e. U. ( R1 " On ) -> ( rank ` d ) = ( rank " ( TC ` d ) ) )
9 7 8 syl 
 |-  ( d e. S -> ( rank ` d ) = ( rank " ( TC ` d ) ) )
10 3 itunitc 
 |-  ( TC ` d ) = U. ran ( U ` d )
11 3 itunifn 
 |-  ( d e. S -> ( U ` d ) Fn _om )
12 fniunfv 
 |-  ( ( U ` d ) Fn _om -> U_ c e. _om ( ( U ` d ) ` c ) = U. ran ( U ` d ) )
13 11 12 syl 
 |-  ( d e. S -> U_ c e. _om ( ( U ` d ) ` c ) = U. ran ( U ` d ) )
14 10 13 eqtr4id 
 |-  ( d e. S -> ( TC ` d ) = U_ c e. _om ( ( U ` d ) ` c ) )
15 14 imaeq2d 
 |-  ( d e. S -> ( rank " ( TC ` d ) ) = ( rank " U_ c e. _om ( ( U ` d ) ` c ) ) )
16 imaiun 
 |-  ( rank " U_ c e. _om ( ( U ` d ) ` c ) ) = U_ c e. _om ( rank " ( ( U ` d ) ` c ) )
17 16 a1i 
 |-  ( d e. S -> ( rank " U_ c e. _om ( ( U ` d ) ` c ) ) = U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) )
18 9 15 17 3eqtrd 
 |-  ( d e. S -> ( rank ` d ) = U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) )
19 dmresi 
 |-  dom ( _I |` U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) ) = U_ c e. _om ( rank " ( ( U ` d ) ` c ) )
20 18 19 eqtr4di 
 |-  ( d e. S -> ( rank ` d ) = dom ( _I |` U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) ) )
21 rankon 
 |-  ( rank ` d ) e. On
22 18 21 eqeltrrdi 
 |-  ( d e. S -> U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) e. On )
23 eloni 
 |-  ( U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) e. On -> Ord U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) )
24 oiid 
 |-  ( 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 ) ) ) )
25 22 23 24 3syl 
 |-  ( d e. S -> OrdIso ( _E , U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) ) = ( _I |` U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) ) )
26 25 dmeqd 
 |-  ( d e. S -> dom OrdIso ( _E , U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) ) = dom ( _I |` U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) ) )
27 20 26 eqtr4d 
 |-  ( d e. S -> ( rank ` d ) = dom OrdIso ( _E , U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) ) )
28 omex 
 |-  _om e. _V
29 wdomref 
 |-  ( _om e. _V -> _om ~<_* _om )
30 28 29 mp1i 
 |-  ( d e. S -> _om ~<_* _om )
31 frfnom 
 |-  ( rec ( ( z e. _V |-> ( har ` ~P ( X X. z ) ) ) , ( har ` ~P X ) ) |` _om ) Fn _om
32 2 fneq1i 
 |-  ( H Fn _om <-> ( rec ( ( z e. _V |-> ( har ` ~P ( X X. z ) ) ) , ( har ` ~P X ) ) |` _om ) Fn _om )
33 31 32 mpbir 
 |-  H Fn _om
34 fniunfv 
 |-  ( H Fn _om -> U_ a e. _om ( H ` a ) = U. ran H )
35 33 34 ax-mp 
 |-  U_ a e. _om ( H ` a ) = U. ran H
36 iunon 
 |-  ( ( _om e. _V /\ A. a e. _om ( H ` a ) e. On ) -> U_ a e. _om ( H ` a ) e. On )
37 28 36 mpan 
 |-  ( A. a e. _om ( H ` a ) e. On -> U_ a e. _om ( H ` a ) e. On )
38 2 hsmexlem9 
 |-  ( a e. _om -> ( H ` a ) e. On )
39 37 38 mprg 
 |-  U_ a e. _om ( H ` a ) e. On
40 35 39 eqeltrri 
 |-  U. ran H e. On
41 40 a1i 
 |-  ( d e. S -> U. ran H e. On )
42 fvssunirn 
 |-  ( H ` c ) C_ U. ran H
43 eqid 
 |-  OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) ) = OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) )
44 1 2 3 4 43 hsmexlem4 
 |-  ( ( c e. _om /\ d e. S ) -> dom OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) ) e. ( H ` c ) )
45 44 ancoms 
 |-  ( ( d e. S /\ c e. _om ) -> dom OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) ) e. ( H ` c ) )
46 42 45 sseldi 
 |-  ( ( d e. S /\ c e. _om ) -> dom OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) ) e. U. ran H )
47 imassrn 
 |-  ( rank " ( ( U ` d ) ` c ) ) C_ ran rank
48 rankf 
 |-  rank : U. ( R1 " On ) --> On
49 frn 
 |-  ( rank : U. ( R1 " On ) --> On -> ran rank C_ On )
50 48 49 ax-mp 
 |-  ran rank C_ On
51 47 50 sstri 
 |-  ( rank " ( ( U ` d ) ` c ) ) C_ On
52 ffun 
 |-  ( rank : U. ( R1 " On ) --> On -> Fun rank )
53 fvex 
 |-  ( ( U ` d ) ` c ) e. _V
54 53 funimaex 
 |-  ( Fun rank -> ( rank " ( ( U ` d ) ` c ) ) e. _V )
55 48 52 54 mp2b 
 |-  ( rank " ( ( U ` d ) ` c ) ) e. _V
56 55 elpw 
 |-  ( ( rank " ( ( U ` d ) ` c ) ) e. ~P On <-> ( rank " ( ( U ` d ) ` c ) ) C_ On )
57 51 56 mpbir 
 |-  ( rank " ( ( U ` d ) ` c ) ) e. ~P On
58 46 57 jctil 
 |-  ( ( 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 
 |-  ( 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 
 |-  OrdIso ( _E , U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) ) = OrdIso ( _E , U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) )
61 43 60 hsmexlem3 
 |-  ( ( ( _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 30 41 59 61 syl21anc 
 |-  ( d e. S -> dom OrdIso ( _E , U_ c e. _om ( rank " ( ( U ` d ) ` c ) ) ) e. ( har ` ~P ( _om X. U. ran H ) ) )
63 27 62 eqeltrd 
 |-  ( d e. S -> ( rank ` d ) e. ( har ` ~P ( _om X. U. ran H ) ) )