Theorem tfrlem10 7483

Description: Lemma for transfinite recursion. We define class C by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to, On. Using this assumption we will prove facts about C that will lead to a contradiction in tfrlem14 7487, thus showing the domain of recs does in fact equal On. Here we show (under the false assumption) that C is a function extending the domain of recs ( F ) by one. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)

Hypotheses

Ref	Expression
tfrlem.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
tfrlem.3	|- C = (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )

Assertion

Ref	Expression
tfrlem10	|- ( dom recs ( F ) e. On -> C Fn suc dom recs ( F ) )

Distinct variable groups:   x, f, y, C   f, F, x, y

Allowed substitution hints:   A( x, y, f)

Proof of Theorem tfrlem10

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . . 6 |- ( F ` recs ( F ) ) e. _V
2		funsng 5937	. . . . . 6 |- ( ( dom recs ( F ) e. On /\ ( F ` recs ( F ) ) e. _V ) -> Fun { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
3	1, 2	mpan2 707	. . . . 5 |- ( dom recs ( F ) e. On -> Fun { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
4		tfrlem.1	. . . . . 6 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
5	4	tfrlem7 7479	. . . . 5 |- Fun recs ( F )
6	3, 5	jctil 560	. . . 4 |- ( dom recs ( F ) e. On -> ( Fun recs ( F ) /\ Fun { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) )
7	1	dmsnop 5609	. . . . . 6 |- dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } = { dom recs ( F ) }
8	7	ineq2i 3811	. . . . 5 |- ( dom recs ( F ) i^i dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = ( dom recs ( F ) i^i { dom recs ( F ) } )
9	4	tfrlem8 7480	. . . . . 6 |- Ord dom recs ( F )
10		orddisj 5762	. . . . . 6 |- ( Ord dom recs ( F ) -> ( dom recs ( F ) i^i { dom recs ( F ) } ) = (/) )
11	9, 10	ax-mp 5	. . . . 5 |- ( dom recs ( F ) i^i { dom recs ( F ) } ) = (/)
12	8, 11	eqtri 2644	. . . 4 |- ( dom recs ( F ) i^i dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = (/)
13		funun 5932	. . . 4 |- ( ( ( Fun recs ( F ) /\ Fun { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) /\ ( dom recs ( F ) i^i dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = (/) ) -> Fun (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) )
14	6, 12, 13	sylancl 694	. . 3 |- ( dom recs ( F ) e. On -> Fun (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) )
15	7	uneq2i 3764	. . . 4 |- ( dom recs ( F ) u. dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = ( dom recs ( F ) u. { dom recs ( F ) } )
16		dmun 5331	. . . 4 |- dom (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = ( dom recs ( F ) u. dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
17		df-suc 5729	. . . 4 |- suc dom recs ( F ) = ( dom recs ( F ) u. { dom recs ( F ) } )
18	15, 16, 17	3eqtr4i 2654	. . 3 |- dom (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = suc dom recs ( F )
19		df-fn 5891	. . 3 |- ( (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) Fn suc dom recs ( F ) <-> ( Fun (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) /\ dom (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = suc dom recs ( F ) ) )
20	14, 18, 19	sylanblrc 697	. 2 |- ( dom recs ( F ) e. On -> (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) Fn suc dom recs ( F ) )
21		tfrlem.3	. . 3 |- C = (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
22	21	fneq1i 5985	. 2 |- ( C Fn suc dom recs ( F ) <-> (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) Fn suc dom recs ( F ) )
23	20, 22	sylibr 224	1 |- ( dom recs ( F ) e. On -> C Fn suc dom recs ( F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   <.cop 4183   dom cdm 5114   |` cres 5116   Ord word 5722   Oncon0 5723   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  recscrecs 7467

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-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-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-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-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-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  tfrlem11  7484  tfrlem12  7485  tfrlem13  7486