Theorem tfrlem14 7487

Description: Lemma for transfinite recursion. Assuming ax-rep 4771, dom recs e. _V <-> recs e. _V, so since dom recs is an ordinal, it must be equal to On. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)

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
tfrlem14	|- dom recs ( F ) = On

Distinct variable group:   x, f, y, F

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

Proof of Theorem tfrlem14

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	tfrlem13 7486	. . 3 |- -. recs ( F ) e. _V
3	1	tfrlem7 7479	. . . 4 |- Fun recs ( F )
4		funex 6482	. . . 4 |- ( ( Fun recs ( F ) /\ dom recs ( F ) e. On ) -> recs ( F ) e. _V )
5	3, 4	mpan 706	. . 3 |- ( dom recs ( F ) e. On -> recs ( F ) e. _V )
6	2, 5	mto 188	. 2 |- -. dom recs ( F ) e. On
7	1	tfrlem8 7480	. . 3 |- Ord dom recs ( F )
8		ordeleqon 6988	. . 3 |- ( Ord dom recs ( F ) <-> ( dom recs ( F ) e. On \/ dom recs ( F ) = On ) )
9	7, 8	mpbi 220	. 2 |- ( dom recs ( F ) e. On \/ dom recs ( F ) = On )
10	6, 9	mtpor 1695	1 |- dom recs ( F ) = On

Syntax hints:   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   dom cdm 5114   |` cres 5116   Ord word 5722   Oncon0 5723   Fun wfun 5882   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-rep 4771  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-reu 2919  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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  tfr1  7493