Theorem tfrlem10 6943

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 6947, 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 5796	. . . . . . 7 |- ( F ` recs ( F ) ) e. _V
2		funsng 5559	. . . . . . 7 |- ( ( dom recs ( F ) e. On /\ ( F ` recs ( F ) ) e. _V ) -> Fun { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
3	1, 2	mpan2 671	. . . . . 6 |- ( dom recs ( F ) e. On -> Fun { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
4		tfrlem.1	. . . . . . 7 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
5	4	tfrlem7 6939	. . . . . 6 |- Fun recs ( F )
6	3, 5	jctil 537	. . . . 5 |- ( dom recs ( F ) e. On -> ( Fun recs ( F ) /\ Fun { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) )
7	1	dmsnop 5408	. . . . . . 7 |- dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } = { dom recs ( F ) }
8	7	ineq2i 3644	. . . . . 6 |- ( dom recs ( F ) i^i dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = ( dom recs ( F ) i^i { dom recs ( F ) } )
9	4	tfrlem8 6940	. . . . . . 7 |- Ord dom recs ( F )
10		orddisj 4852	. . . . . . 7 |- ( Ord dom recs ( F ) -> ( dom recs ( F ) i^i { dom recs ( F ) } ) = (/) )
11	9, 10	ax-mp 5	. . . . . 6 |- ( dom recs ( F ) i^i { dom recs ( F ) } ) = (/)
12	8, 11	eqtri 2479	. . . . 5 |- ( dom recs ( F ) i^i dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = (/)
13		funun 5555	. . . . 5 |- ( ( ( 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 662	. . . 4 |- ( dom recs ( F ) e. On -> Fun (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) )
15	7	uneq2i 3602	. . . . 5 |- ( dom recs ( F ) u. dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = ( dom recs ( F ) u. { dom recs ( F ) } )
16		dmun 5141	. . . . 5 |- dom (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = ( dom recs ( F ) u. dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
17		df-suc 4820	. . . . 5 |- suc dom recs ( F ) = ( dom recs ( F ) u. { dom recs ( F ) } )
18	15, 16, 17	3eqtr4i 2489	. . . 4 |- dom (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = suc dom recs ( F )
19	14, 18	jctir 538	. . 3 |- ( dom recs ( F ) e. On -> ( 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		df-fn 5516	. . 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 ) ) )
21	19, 20	sylibr 212	. 2 |- ( dom recs ( F ) e. On -> (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) Fn suc dom recs ( F ) )
22		tfrlem.3	. . 3 |- C = (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
23	22	fneq1i 5600	. 2 |- ( C Fn suc dom recs ( F ) <-> (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) Fn suc dom recs ( F ) )
24	21, 23	sylibr 212	1 |- ( dom recs ( F ) e. On -> C Fn suc dom recs ( F ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   {cab 2436   A.wral 2793   E.wrex 2794   _Vcvv 3065   u. cun 3421   i^i cin 3422   (/)c0 3732   {csn 3972   <.cop 3978   Ord word 4813   Oncon0 4814   suc csuc 4816   dom cdm 4935   |` cres 4937   Fun wfun 5507   Fn wfn 5508   ` cfv 5513  recscrecs 6928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-fv 5521  df-recs 6929

This theorem is referenced by:  tfrlem11  6944  tfrlem12  6945  tfrlem13  6946