Theorem tfrlem13 7486

Description: Lemma for transfinite recursion. If recs is a set function, then C is acceptable, and thus a subset of recs, but dom C is bigger than dom recs. This is a contradiction, so recs must be a proper class function. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2014.)

Hypothesis

Ref	Expression
tfrlem.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }

Assertion

Ref	Expression
tfrlem13	|- -. recs ( F ) e. _V

Distinct variable group:   x, f, y, F

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

Proof of Theorem tfrlem13

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . 4 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2	1	tfrlem8 7480	. . 3 |- Ord dom recs ( F )
3		ordirr 5741	. . 3 |- ( Ord dom recs ( F ) -> -. dom recs ( F ) e. dom recs ( F ) )
4	2, 3	ax-mp 5	. 2 |- -. dom recs ( F ) e. dom recs ( F )
5		eqid 2622	. . . . 5 |- (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
6	1, 5	tfrlem12 7485	. . . 4 |- (recs ( F ) e. _V -> (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) e. A )
7		elssuni 4467	. . . . 5 |- ( (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) e. A -> (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) C_ U. A )
8	1	recsfval 7477	. . . . 5 |- recs ( F ) = U. A
9	7, 8	syl6sseqr 3652	. . . 4 |- ( (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) e. A -> (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) C_ recs ( F ) )
10		dmss 5323	. . . 4 |- ( (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) C_ recs ( F ) -> dom (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) C_ dom recs ( F ) )
11	6, 9, 10	3syl 18	. . 3 |- (recs ( F ) e. _V -> dom (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) C_ dom recs ( F ) )
12	2	a1i 11	. . . . . 6 |- (recs ( F ) e. _V -> Ord dom recs ( F ) )
13		dmexg 7097	. . . . . 6 |- (recs ( F ) e. _V -> dom recs ( F ) e. _V )
14		elon2 5734	. . . . . 6 |- ( dom recs ( F ) e. On <-> ( Ord dom recs ( F ) /\ dom recs ( F ) e. _V ) )
15	12, 13, 14	sylanbrc 698	. . . . 5 |- (recs ( F ) e. _V -> dom recs ( F ) e. On )
16		sucidg 5803	. . . . 5 |- ( dom recs ( F ) e. On -> dom recs ( F ) e. suc dom recs ( F ) )
17	15, 16	syl 17	. . . 4 |- (recs ( F ) e. _V -> dom recs ( F ) e. suc dom recs ( F ) )
18	1, 5	tfrlem10 7483	. . . . 5 |- ( dom recs ( F ) e. On -> (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) Fn suc dom recs ( F ) )
19		fndm 5990	. . . . 5 |- ( (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) Fn suc dom recs ( F ) -> dom (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = suc dom recs ( F ) )
20	15, 18, 19	3syl 18	. . . 4 |- (recs ( F ) e. _V -> dom (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) = suc dom recs ( F ) )
21	17, 20	eleqtrrd 2704	. . 3 |- (recs ( F ) e. _V -> dom recs ( F ) e. dom (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) )
22	11, 21	sseldd 3604	. 2 |- (recs ( F ) e. _V -> dom recs ( F ) e. dom recs ( F ) )
23	4, 22	mto 188	1 |- -. recs ( F ) e. _V

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   u. cun 3572   C_ wss 3574   {csn 4177   <.cop 4183   U.cuni 4436   dom cdm 5114   |` cres 5116   Ord word 5722   Oncon0 5723   suc csuc 5725   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:  tfrlem14  7487  tfrlem15  7488  tfrlem16  7489  tfr2b  7492