Theorem tfrlem12 7485

Description: Lemma for transfinite recursion. Show C is an acceptable function. (Contributed by NM, 15-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
tfrlem12	|- (recs ( F ) e. _V -> C e. A )

Distinct variable groups:   x, f, y, C   f, F, x, y

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

Proof of Theorem tfrlem12

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . . . 6 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2	1	tfrlem8 7480	. . . . 5 |- Ord dom recs ( F )
3	2	a1i 11	. . . 4 |- (recs ( F ) e. _V -> Ord dom recs ( F ) )
4		dmexg 7097	. . . 4 |- (recs ( F ) e. _V -> dom recs ( F ) e. _V )
5		elon2 5734	. . . 4 |- ( dom recs ( F ) e. On <-> ( Ord dom recs ( F ) /\ dom recs ( F ) e. _V ) )
6	3, 4, 5	sylanbrc 698	. . 3 |- (recs ( F ) e. _V -> dom recs ( F ) e. On )
7		suceloni 7013	. . . 4 |- ( dom recs ( F ) e. On -> suc dom recs ( F ) e. On )
8		tfrlem.3	. . . . 5 |- C = (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
9	1, 8	tfrlem10 7483	. . . 4 |- ( dom recs ( F ) e. On -> C Fn suc dom recs ( F ) )
10	1, 8	tfrlem11 7484	. . . . . 6 |- ( dom recs ( F ) e. On -> ( z e. suc dom recs ( F ) -> ( C ` z ) = ( F ` ( C |` z ) ) ) )
11	10	ralrimiv 2965	. . . . 5 |- ( dom recs ( F ) e. On -> A. z e. suc dom recs ( F ) ( C ` z ) = ( F ` ( C |` z ) ) )
12		fveq2 6191	. . . . . . 7 |- ( z = y -> ( C ` z ) = ( C ` y ) )
13		reseq2 5391	. . . . . . . 8 |- ( z = y -> ( C |` z ) = ( C |` y ) )
14	13	fveq2d 6195	. . . . . . 7 |- ( z = y -> ( F ` ( C |` z ) ) = ( F ` ( C |` y ) ) )
15	12, 14	eqeq12d 2637	. . . . . 6 |- ( z = y -> ( ( C ` z ) = ( F ` ( C |` z ) ) <-> ( C ` y ) = ( F ` ( C |` y ) ) ) )
16	15	cbvralv 3171	. . . . 5 |- ( A. z e. suc dom recs ( F ) ( C ` z ) = ( F ` ( C |` z ) ) <-> A. y e. suc dom recs ( F ) ( C ` y ) = ( F ` ( C |` y ) ) )
17	11, 16	sylib 208	. . . 4 |- ( dom recs ( F ) e. On -> A. y e. suc dom recs ( F ) ( C ` y ) = ( F ` ( C |` y ) ) )
18		fneq2 5980	. . . . . 6 |- ( x = suc dom recs ( F ) -> ( C Fn x <-> C Fn suc dom recs ( F ) ) )
19		raleq 3138	. . . . . 6 |- ( x = suc dom recs ( F ) -> ( A. y e. x ( C ` y ) = ( F ` ( C |` y ) ) <-> A. y e. suc dom recs ( F ) ( C ` y ) = ( F ` ( C |` y ) ) ) )
20	18, 19	anbi12d 747	. . . . 5 |- ( x = suc dom recs ( F ) -> ( ( C Fn x /\ A. y e. x ( C ` y ) = ( F ` ( C |` y ) ) ) <-> ( C Fn suc dom recs ( F ) /\ A. y e. suc dom recs ( F ) ( C ` y ) = ( F ` ( C |` y ) ) ) ) )
21	20	rspcev 3309	. . . 4 |- ( ( suc dom recs ( F ) e. On /\ ( C Fn suc dom recs ( F ) /\ A. y e. suc dom recs ( F ) ( C ` y ) = ( F ` ( C |` y ) ) ) ) -> E. x e. On ( C Fn x /\ A. y e. x ( C ` y ) = ( F ` ( C |` y ) ) ) )
22	7, 9, 17, 21	syl12anc 1324	. . 3 |- ( dom recs ( F ) e. On -> E. x e. On ( C Fn x /\ A. y e. x ( C ` y ) = ( F ` ( C |` y ) ) ) )
23	6, 22	syl 17	. 2 |- (recs ( F ) e. _V -> E. x e. On ( C Fn x /\ A. y e. x ( C ` y ) = ( F ` ( C |` y ) ) ) )
24		snex 4908	. . . . 5 |- { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } e. _V
25		unexg 6959	. . . . 5 |- ( (recs ( F ) e. _V /\ { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } e. _V ) -> (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) e. _V )
26	24, 25	mpan2 707	. . . 4 |- (recs ( F ) e. _V -> (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) e. _V )
27	8, 26	syl5eqel 2705	. . 3 |- (recs ( F ) e. _V -> C e. _V )
28		fneq1 5979	. . . . . 6 |- ( f = C -> ( f Fn x <-> C Fn x ) )
29		fveq1 6190	. . . . . . . 8 |- ( f = C -> ( f ` y ) = ( C ` y ) )
30		reseq1 5390	. . . . . . . . 9 |- ( f = C -> ( f |` y ) = ( C |` y ) )
31	30	fveq2d 6195	. . . . . . . 8 |- ( f = C -> ( F ` ( f |` y ) ) = ( F ` ( C |` y ) ) )
32	29, 31	eqeq12d 2637	. . . . . . 7 |- ( f = C -> ( ( f ` y ) = ( F ` ( f |` y ) ) <-> ( C ` y ) = ( F ` ( C |` y ) ) ) )
33	32	ralbidv 2986	. . . . . 6 |- ( f = C -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) <-> A. y e. x ( C ` y ) = ( F ` ( C |` y ) ) ) )
34	28, 33	anbi12d 747	. . . . 5 |- ( f = C -> ( ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) <-> ( C Fn x /\ A. y e. x ( C ` y ) = ( F ` ( C |` y ) ) ) ) )
35	34	rexbidv 3052	. . . 4 |- ( f = C -> ( E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) <-> E. x e. On ( C Fn x /\ A. y e. x ( C ` y ) = ( F ` ( C |` y ) ) ) ) )
36	35, 1	elab2g 3353	. . 3 |- ( C e. _V -> ( C e. A <-> E. x e. On ( C Fn x /\ A. y e. x ( C ` y ) = ( F ` ( C |` y ) ) ) ) )
37	27, 36	syl 17	. 2 |- (recs ( F ) e. _V -> ( C e. A <-> E. x e. On ( C Fn x /\ A. y e. x ( C ` y ) = ( F ` ( C |` y ) ) ) ) )
38	23, 37	mpbird 247	1 |- (recs ( F ) e. _V -> C e. A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   u. cun 3572   {csn 4177   <.cop 4183   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:  tfrlem13  7486