Theorem harwdom 8495

Description: The Hartogs function is weakly dominated by ~P ( X X. X ). This follows from a more precise analysis of the bound used in hartogs 8449 to prove that (har ` X ) is a set. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
harwdom	|- ( X e. V -> (har ` X ) ~<_* ~P ( X X. X ) )

Proof of Theorem harwdom

Dummy variables y r f s t w x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } = { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) }
2		eqid 2622	. . . . . 6 |- { <. s , t >. | E. w e. y E. z e. y ( ( s = ( f ` w ) /\ t = ( f ` z ) ) /\ w _E z ) } = { <. s , t >. | E. w e. y E. z e. y ( ( s = ( f ` w ) /\ t = ( f ` z ) ) /\ w _E z ) }
3	1, 2	hartogslem1 8447	. . . . 5 |- ( dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } C_ ~P ( X X. X ) /\ Fun { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } /\ ( X e. V -> ran { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } = { x e. On | x ~<_ X } ) )
4	3	simp2i 1071	. . . 4 |- Fun { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) }
5	3	simp1i 1070	. . . . 5 |- dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } C_ ~P ( X X. X )
6		sqxpexg 6963	. . . . . 6 |- ( X e. V -> ( X X. X ) e. _V )
7		pwexg 4850	. . . . . 6 |- ( ( X X. X ) e. _V -> ~P ( X X. X ) e. _V )
8	6, 7	syl 17	. . . . 5 |- ( X e. V -> ~P ( X X. X ) e. _V )
9		ssexg 4804	. . . . 5 |- ( ( dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } C_ ~P ( X X. X ) /\ ~P ( X X. X ) e. _V ) -> dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } e. _V )
10	5, 8, 9	sylancr 695	. . . 4 |- ( X e. V -> dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } e. _V )
11		funex 6482	. . . 4 |- ( ( Fun { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } /\ dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } e. _V ) -> { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } e. _V )
12	4, 10, 11	sylancr 695	. . 3 |- ( X e. V -> { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } e. _V )
13		funfn 5918	. . . . . 6 |- ( Fun { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } <-> { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } Fn dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } )
14	4, 13	mpbi 220	. . . . 5 |- { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } Fn dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) }
15	14	a1i 11	. . . 4 |- ( X e. V -> { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } Fn dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } )
16	3	simp3i 1072	. . . . 5 |- ( X e. V -> ran { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } = { x e. On | x ~<_ X } )
17		harval 8467	. . . . 5 |- ( X e. V -> (har ` X ) = { x e. On | x ~<_ X } )
18	16, 17	eqtr4d 2659	. . . 4 |- ( X e. V -> ran { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } = (har ` X ) )
19		df-fo 5894	. . . 4 |- ( { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } : dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } -onto-> (har ` X ) <-> ( { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } Fn dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } /\ ran { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } = (har ` X ) ) )
20	15, 18, 19	sylanbrc 698	. . 3 |- ( X e. V -> { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } : dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } -onto-> (har ` X ) )
21		fowdom 8476	. . 3 |- ( ( { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } e. _V /\ { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } : dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } -onto-> (har ` X ) ) -> (har ` X ) ~<_* dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } )
22	12, 20, 21	syl2anc 693	. 2 |- ( X e. V -> (har ` X ) ~<_* dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } )
23		ssdomg 8001	. . . 4 |- ( ~P ( X X. X ) e. _V -> ( dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } C_ ~P ( X X. X ) -> dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } ~<_ ~P ( X X. X ) ) )
24	8, 5, 23	mpisyl 21	. . 3 |- ( X e. V -> dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } ~<_ ~P ( X X. X ) )
25		domwdom 8479	. . 3 |- ( dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } ~<_ ~P ( X X. X ) -> dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } ~<_* ~P ( X X. X ) )
26	24, 25	syl 17	. 2 |- ( X e. V -> dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } ~<_* ~P ( X X. X ) )
27		wdomtr 8480	. 2 |- ( ( (har ` X ) ~<_* dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } /\ dom { <. r , y >. | ( ( ( dom r C_ X /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\ ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } ~<_* ~P ( X X. X ) ) -> (har ` X ) ~<_* ~P ( X X. X ) )
28	22, 26, 27	syl2anc 693	1 |- ( X e. V -> (har ` X ) ~<_* ~P ( X X. X ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   {copab 4712   _I cid 5023   _E cep 5028   We wwe 5072   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888   ~<_ cdom 7953  OrdIsocoi 8414  harchar 8461   ~<_^* cwdom 8462

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

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-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-wrecs 7407  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-oi 8415  df-har 8463  df-wdom 8464

This theorem is referenced by:  gchhar  9501