Theorem tfrlem10 7074

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 7078, 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 5882	. . . . . . 7 |- ( F ` recs ( F ) ) e. _V
2		funsng 5640	. . . . . . 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 7070	. . . . . 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 5488	. . . . . . 7 |- dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } = { dom recs ( F ) }
8	7	ineq2i 3693	. . . . . 6 |- ( dom recs ( F ) i^i dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = ( dom recs ( F ) i^i { dom recs ( F ) } )
9	4	tfrlem8 7071	. . . . . . 7 |- Ord dom recs ( F )
10		orddisj 4925	. . . . . . 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 2486	. . . . 5 |- ( dom recs ( F ) i^i dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = (/)
13		funun 5636	. . . . 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 3651	. . . . 5 |- ( dom recs ( F ) u. dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = ( dom recs ( F ) u. { dom recs ( F ) } )
16		dmun 5219	. . . . 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 4893	. . . . 5 |- suc dom recs ( F ) = ( dom recs ( F ) u. { dom recs ( F ) } )
18	15, 16, 17	3eqtr4i 2496	. . . 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 5597	. . 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 5681	. 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 1395   e. wcel 1819   {cab 2442   A.wral 2807   E.wrex 2808   _Vcvv 3109   u. cun 3469   i^i cin 3470   (/)c0 3793   {csn 4032   <.cop 4038   Ord word 4886   Oncon0 4887   suc csuc 4889   dom cdm 5008   |` cres 5010   Fun wfun 5588   Fn wfn 5589   ` cfv 5594  recscrecs 7059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-recs 7060

This theorem is referenced by:  tfrlem11  7075  tfrlem12  7076  tfrlem13  7077