Theorem rdglim2 7528

Description: The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values. (Contributed by NM, 23-Apr-1995.)

Assertion

Ref	Expression
rdglim2	|- ( ( B e. C /\ Lim B ) -> ( rec ( F , A ) ` B ) = U. { y | E. x e. B y = ( rec ( F , A ) ` x ) } )

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

Allowed substitution hints:   C( x, y)

Proof of Theorem rdglim2

Step	Hyp	Ref	Expression
1		rdglim 7522	. 2 |- ( ( B e. C /\ Lim B ) -> ( rec ( F , A ) ` B ) = U. ( rec ( F , A ) " B ) )
2		dfima3 5469	. . . . 5 |- ( rec ( F , A ) " B ) = { y | E. x ( x e. B /\ <. x , y >. e. rec ( F , A ) ) }
3		df-rex 2918	. . . . . . 7 |- ( E. x e. B y = ( rec ( F , A ) ` x ) <-> E. x ( x e. B /\ y = ( rec ( F , A ) ` x ) ) )
4		limord 5784	. . . . . . . . . . 11 |- ( Lim B -> Ord B )
5		ordelord 5745	. . . . . . . . . . . . 13 |- ( ( Ord B /\ x e. B ) -> Ord x )
6	5	ex 450	. . . . . . . . . . . 12 |- ( Ord B -> ( x e. B -> Ord x ) )
7		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
8	7	elon 5732	. . . . . . . . . . . 12 |- ( x e. On <-> Ord x )
9	6, 8	syl6ibr 242	. . . . . . . . . . 11 |- ( Ord B -> ( x e. B -> x e. On ) )
10	4, 9	syl 17	. . . . . . . . . 10 |- ( Lim B -> ( x e. B -> x e. On ) )
11		eqcom 2629	. . . . . . . . . . 11 |- ( y = ( rec ( F , A ) ` x ) <-> ( rec ( F , A ) ` x ) = y )
12		rdgfnon 7514	. . . . . . . . . . . 12 |- rec ( F , A ) Fn On
13		fnopfvb 6237	. . . . . . . . . . . 12 |- ( ( rec ( F , A ) Fn On /\ x e. On ) -> ( ( rec ( F , A ) ` x ) = y <-> <. x , y >. e. rec ( F , A ) ) )
14	12, 13	mpan 706	. . . . . . . . . . 11 |- ( x e. On -> ( ( rec ( F , A ) ` x ) = y <-> <. x , y >. e. rec ( F , A ) ) )
15	11, 14	syl5bb 272	. . . . . . . . . 10 |- ( x e. On -> ( y = ( rec ( F , A ) ` x ) <-> <. x , y >. e. rec ( F , A ) ) )
16	10, 15	syl6 35	. . . . . . . . 9 |- ( Lim B -> ( x e. B -> ( y = ( rec ( F , A ) ` x ) <-> <. x , y >. e. rec ( F , A ) ) ) )
17	16	pm5.32d 671	. . . . . . . 8 |- ( Lim B -> ( ( x e. B /\ y = ( rec ( F , A ) ` x ) ) <-> ( x e. B /\ <. x , y >. e. rec ( F , A ) ) ) )
18	17	exbidv 1850	. . . . . . 7 |- ( Lim B -> ( E. x ( x e. B /\ y = ( rec ( F , A ) ` x ) ) <-> E. x ( x e. B /\ <. x , y >. e. rec ( F , A ) ) ) )
19	3, 18	syl5rbb 273	. . . . . 6 |- ( Lim B -> ( E. x ( x e. B /\ <. x , y >. e. rec ( F , A ) ) <-> E. x e. B y = ( rec ( F , A ) ` x ) ) )
20	19	abbidv 2741	. . . . 5 |- ( Lim B -> { y | E. x ( x e. B /\ <. x , y >. e. rec ( F , A ) ) } = { y | E. x e. B y = ( rec ( F , A ) ` x ) } )
21	2, 20	syl5eq 2668	. . . 4 |- ( Lim B -> ( rec ( F , A ) " B ) = { y | E. x e. B y = ( rec ( F , A ) ` x ) } )
22	21	unieqd 4446	. . 3 |- ( Lim B -> U. ( rec ( F , A ) " B ) = U. { y | E. x e. B y = ( rec ( F , A ) ` x ) } )
23	22	adantl 482	. 2 |- ( ( B e. C /\ Lim B ) -> U. ( rec ( F , A ) " B ) = U. { y | E. x e. B y = ( rec ( F , A ) ` x ) } )
24	1, 23	eqtrd 2656	1 |- ( ( B e. C /\ Lim B ) -> ( rec ( F , A ) ` B ) = U. { y | E. x e. B y = ( rec ( F , A ) ` x ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   <.cop 4183   U.cuni 4436   "cima 5117   Ord word 5722   Oncon0 5723   Lim wlim 5724   Fn wfn 5883   ` cfv 5888   reccrdg 7505

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-pw 4160  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-lim 5728  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  df-rdg 7506

This theorem is referenced by:  rdglim2a  7529