Theorem tfrlem15 7488

Description: Lemma for transfinite recursion. Without assuming ax-rep 4771, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed 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
tfrlem15	|- ( B e. On -> ( B e. dom recs ( F ) <-> (recs ( F ) |` B ) e. _V ) )

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

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

Proof of Theorem tfrlem15

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	tfrlem9a 7482	. . 3 |- ( B e. dom recs ( F ) -> (recs ( F ) |` B ) e. _V )
3	2	adantl 482	. 2 |- ( ( B e. On /\ B e. dom recs ( F ) ) -> (recs ( F ) |` B ) e. _V )
4	1	tfrlem13 7486	. . . 4 |- -. recs ( F ) e. _V
5		simpr 477	. . . . 5 |- ( ( B e. On /\ (recs ( F ) |` B ) e. _V ) -> (recs ( F ) |` B ) e. _V )
6		resss 5422	. . . . . . . 8 |- (recs ( F ) |` B ) C_ recs ( F )
7	6	a1i 11	. . . . . . 7 |- ( dom recs ( F ) C_ B -> (recs ( F ) |` B ) C_ recs ( F ) )
8	1	tfrlem6 7478	. . . . . . . . 9 |- Rel recs ( F )
9		resdm 5441	. . . . . . . . 9 |- ( Rel recs ( F ) -> (recs ( F ) |` dom recs ( F ) ) = recs ( F ) )
10	8, 9	ax-mp 5	. . . . . . . 8 |- (recs ( F ) |` dom recs ( F ) ) = recs ( F )
11		ssres2 5425	. . . . . . . 8 |- ( dom recs ( F ) C_ B -> (recs ( F ) |` dom recs ( F ) ) C_ (recs ( F ) |` B ) )
12	10, 11	syl5eqssr 3650	. . . . . . 7 |- ( dom recs ( F ) C_ B -> recs ( F ) C_ (recs ( F ) |` B ) )
13	7, 12	eqssd 3620	. . . . . 6 |- ( dom recs ( F ) C_ B -> (recs ( F ) |` B ) = recs ( F ) )
14	13	eleq1d 2686	. . . . 5 |- ( dom recs ( F ) C_ B -> ( (recs ( F ) |` B ) e. _V <-> recs ( F ) e. _V ) )
15	5, 14	syl5ibcom 235	. . . 4 |- ( ( B e. On /\ (recs ( F ) |` B ) e. _V ) -> ( dom recs ( F ) C_ B -> recs ( F ) e. _V ) )
16	4, 15	mtoi 190	. . 3 |- ( ( B e. On /\ (recs ( F ) |` B ) e. _V ) -> -. dom recs ( F ) C_ B )
17	1	tfrlem8 7480	. . . 4 |- Ord dom recs ( F )
18		eloni 5733	. . . . 5 |- ( B e. On -> Ord B )
19	18	adantr 481	. . . 4 |- ( ( B e. On /\ (recs ( F ) |` B ) e. _V ) -> Ord B )
20		ordtri1 5756	. . . . 5 |- ( ( Ord dom recs ( F ) /\ Ord B ) -> ( dom recs ( F ) C_ B <-> -. B e. dom recs ( F ) ) )
21	20	con2bid 344	. . . 4 |- ( ( Ord dom recs ( F ) /\ Ord B ) -> ( B e. dom recs ( F ) <-> -. dom recs ( F ) C_ B ) )
22	17, 19, 21	sylancr 695	. . 3 |- ( ( B e. On /\ (recs ( F ) |` B ) e. _V ) -> ( B e. dom recs ( F ) <-> -. dom recs ( F ) C_ B ) )
23	16, 22	mpbird 247	. 2 |- ( ( B e. On /\ (recs ( F ) |` B ) e. _V ) -> B e. dom recs ( F ) )
24	3, 23	impbida 877	1 |- ( B e. On -> ( B e. dom recs ( F ) <-> (recs ( F ) |` B ) e. _V ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   dom cdm 5114   |` cres 5116   Rel wrel 5119   Ord word 5722   Oncon0 5723   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:  tfrlem16  7489  tfr2b  7492