Theorem tfrlem10 7046

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 7050, 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 5867	. . . . . . 7 |- ( F ` recs ( F ) ) e. _V
2		funsng 5625	. . . . . . 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 7042	. . . . . 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 5473	. . . . . . 7 |- dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } = { dom recs ( F ) }
8	7	ineq2i 3690	. . . . . 6 |- ( dom recs ( F ) i^i dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = ( dom recs ( F ) i^i { dom recs ( F ) } )
9	4	tfrlem8 7043	. . . . . . 7 |- Ord dom recs ( F )
10		orddisj 4909	. . . . . . 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 2489	. . . . 5 |- ( dom recs ( F ) i^i dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = (/)
13		funun 5621	. . . . 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 3648	. . . . 5 |- ( dom recs ( F ) u. dom { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = ( dom recs ( F ) u. { dom recs ( F ) } )
16		dmun 5200	. . . . 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 4877	. . . . 5 |- suc dom recs ( F ) = ( dom recs ( F ) u. { dom recs ( F ) } )
18	15, 16, 17	3eqtr4i 2499	. . . 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 5582	. . 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 5666	. 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 1374   e. wcel 1762   {cab 2445   A.wral 2807   E.wrex 2808   _Vcvv 3106   u. cun 3467   i^i cin 3468   (/)c0 3778   {csn 4020   <.cop 4026   Ord word 4870   Oncon0 4871   suc csuc 4873   dom cdm 4992   |` cres 4994   Fun wfun 5573   Fn wfn 5574   ` cfv 5579  recscrecs 7031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-fv 5587  df-recs 7032

This theorem is referenced by:  tfrlem11  7047  tfrlem12  7048  tfrlem13  7049