Theorem hsmex 9254

Description: The collection of hereditarily size-limited well-founded sets comprise a set. The proof is that of Randall Holmes at http://math.boisestate.edu/~holmes/holmes/hereditary.pdf, with modifications to use Hartogs' theorem instead of the weak variant (inconsequentially weakening some intermediate results), and making the well-foundedness condition explicit to avoid a direct dependence on ax-reg 8497. (Contributed by Stefan O'Rear, 14-Feb-2015.)

Assertion

Ref	Expression
hsmex	|- ( X e. V -> { s e. U. ( R1 " On ) | A. x e. ( TC ` { s } ) x ~<_ X } e. _V )

Distinct variable group:   x, s, X

Allowed substitution hints:   V( x, s)

Proof of Theorem hsmex

Dummy variables a b c d e f y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4657	. . . . 5 |- ( a = X -> ( x ~<_ a <-> x ~<_ X ) )
2	1	ralbidv 2986	. . . 4 |- ( a = X -> ( A. x e. ( TC ` { s } ) x ~<_ a <-> A. x e. ( TC ` { s } ) x ~<_ X ) )
3	2	rabbidv 3189	. . 3 |- ( a = X -> { s e. U. ( R1 " On ) | A. x e. ( TC ` { s } ) x ~<_ a } = { s e. U. ( R1 " On ) | A. x e. ( TC ` { s } ) x ~<_ X } )
4	3	eleq1d 2686	. 2 |- ( a = X -> ( { s e. U. ( R1 " On ) | A. x e. ( TC ` { s } ) x ~<_ a } e. _V <-> { s e. U. ( R1 " On ) | A. x e. ( TC ` { s } ) x ~<_ X } e. _V ) )
5		vex 3203	. . 3 |- a e. _V
6		eqid 2622	. . 3 |- ( rec ( ( d e. _V |-> (har ` ~P ( a X. d ) ) ) , (har ` ~P a ) ) |` om ) = ( rec ( ( d e. _V |-> (har ` ~P ( a X. d ) ) ) , (har ` ~P a ) ) |` om )
7		rdgeq2 7508	. . . . . 6 |- ( e = b -> rec ( ( f e. _V |-> U. f ) , e ) = rec ( ( f e. _V |-> U. f ) , b ) )
8		unieq 4444	. . . . . . . 8 |- ( f = c -> U. f = U. c )
9	8	cbvmptv 4750	. . . . . . 7 |- ( f e. _V |-> U. f ) = ( c e. _V |-> U. c )
10		rdgeq1 7507	. . . . . . 7 |- ( ( f e. _V |-> U. f ) = ( c e. _V |-> U. c ) -> rec ( ( f e. _V |-> U. f ) , b ) = rec ( ( c e. _V |-> U. c ) , b ) )
11	9, 10	ax-mp 5	. . . . . 6 |- rec ( ( f e. _V |-> U. f ) , b ) = rec ( ( c e. _V |-> U. c ) , b )
12	7, 11	syl6eq 2672	. . . . 5 |- ( e = b -> rec ( ( f e. _V |-> U. f ) , e ) = rec ( ( c e. _V |-> U. c ) , b ) )
13	12	reseq1d 5395	. . . 4 |- ( e = b -> ( rec ( ( f e. _V |-> U. f ) , e ) |` om ) = ( rec ( ( c e. _V |-> U. c ) , b ) |` om ) )
14	13	cbvmptv 4750	. . 3 |- ( e e. _V |-> ( rec ( ( f e. _V |-> U. f ) , e ) |` om ) ) = ( b e. _V |-> ( rec ( ( c e. _V |-> U. c ) , b ) |` om ) )
15		eqid 2622	. . 3 |- { s e. U. ( R1 " On ) | A. x e. ( TC ` { s } ) x ~<_ a } = { s e. U. ( R1 " On ) | A. x e. ( TC ` { s } ) x ~<_ a }
16		eqid 2622	. . 3 |- OrdIso ( _E , ( rank " ( ( ( e e. _V |-> ( rec ( ( f e. _V |-> U. f ) , e ) |` om ) ) ` z ) ` y ) ) ) = OrdIso ( _E , ( rank " ( ( ( e e. _V |-> ( rec ( ( f e. _V |-> U. f ) , e ) |` om ) ) ` z ) ` y ) ) )
17	5, 6, 14, 15, 16	hsmexlem6 9253	. 2 |- { s e. U. ( R1 " On ) | A. x e. ( TC ` { s } ) x ~<_ a } e. _V
18	4, 17	vtoclg 3266	1 |- ( X e. V -> { s e. U. ( R1 " On ) | A. x e. ( TC ` { s } ) x ~<_ X } e. _V )

Syntax hints:   -> wi 4   = 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   |` cres 5116   "cima 5117   Oncon0 5723   ` 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:  hsmex2  9255