Theorem tfrlem10 6607

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 6611, 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 5701	. . . . . . 7 |- ( F ` recs ( F ) ) e. _V
2		funsng 5456	. . . . . . 7 |- ( ( dom recs ( F ) e. On /\ ( F ` recs ( F ) ) e. _V ) -> Fun { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
3	1, 2	mpan2 653	. . . . . 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 6603	. . . . . 6 |- Fun recs ( F )
6	3, 5	jctil 524	. . . . 5 |- ( dom recs ( F ) e. On -> ( Fun recs ( F ) /\ Fun { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) )
7	1	dmsnop 5303	. . . . . . 7 |- dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } = { dom recs ( F ) }
8	7	ineq2i 3499	. . . . . 6 |- ( dom recs ( F ) i^i dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = ( dom recs ( F ) i^i { dom recs ( F ) } )
9	4	tfrlem8 6604	. . . . . . 7 |- Ord dom recs ( F )
10		orddisj 4579	. . . . . . 7 |- ( Ord dom recs ( F ) -> ( dom recs ( F ) i^i { dom recs ( F ) } ) = (/) )
11	9, 10	ax-mp 8	. . . . . 6 |- ( dom recs ( F ) i^i { dom recs ( F ) } ) = (/)
12	8, 11	eqtri 2424	. . . . 5 |- ( dom recs ( F ) i^i dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = (/)
13		funun 5454	. . . . 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 644	. . . 4 |- ( dom recs ( F ) e. On -> Fun (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) )
15	7	uneq2i 3458	. . . . 5 |- ( dom recs ( F ) u. dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = ( dom recs ( F ) u. { dom recs ( F ) } )
16		dmun 5035	. . . . 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 4547	. . . . 5 |- suc dom recs ( F ) = ( dom recs ( F ) u. { dom recs ( F ) } )
18	15, 16, 17	3eqtr4i 2434	. . . 4 |- dom (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = suc dom recs ( F )
19	14, 18	jctir 525	. . 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 5416	. . 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 204	. 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 5498	. 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 204	1 |- ( dom recs ( F ) e. On -> C Fn suc dom recs ( F ) )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667   _Vcvv 2916   u. cun 3278   i^i cin 3279   (/)c0 3588   {csn 3774   <.cop 3777   Ord word 4540   Oncon0 4541   suc csuc 4543   dom cdm 4837   |` cres 4839   Fun wfun 5407   Fn wfn 5408   ` cfv 5413  recscrecs 6591

This theorem is referenced by:  tfrlem11  6608  tfrlem12  6609  tfrlem13  6610

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-recs 6592