Theorem aomclem2 37625

Description: Lemma for dfac11 37632. Successor case 2, a choice function for subsets of ( R1 ` dom z ). (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypotheses

Ref	Expression
aomclem2.b	|- B = { <. a , b >. | E. c e. ( R1 ` U. dom z ) ( ( c e. b /\ -. c e. a ) /\ A. d e. ( R1 ` U. dom z ) ( d ( z ` U. dom z ) c -> ( d e. a <-> d e. b ) ) ) }
aomclem2.c	|- C = ( a e. _V |-> sup ( ( y ` a ) , ( R1 ` dom z ) , B ) )
aomclem2.on	|- ( ph -> dom z e. On )
aomclem2.su	|- ( ph -> dom z = suc U. dom z )
aomclem2.we	|- ( ph -> A. a e. dom z ( z ` a ) We ( R1 ` a ) )
aomclem2.a	|- ( ph -> A e. On )
aomclem2.za	|- ( ph -> dom z C_ A )
aomclem2.y	|- ( ph -> A. a e. ~P ( R1 ` A ) ( a =/= (/) -> ( y ` a ) e. ( ( ~P a i^i Fin ) \ { (/) } ) ) )

Assertion

Ref	Expression
aomclem2	|- ( ph -> A. a e. ~P ( R1 ` dom z ) ( a =/= (/) -> ( C ` a ) e. a ) )

Distinct variable groups:   y, z, a, b, c, d   ph, a

Allowed substitution hints:   ph( y, z, b, c, d)   A( y, z, a, b, c, d)   B( y, z, a, b, c, d)   C( y, z, a, b, c, d)

Proof of Theorem aomclem2

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- a e. _V
2		aomclem2.y	. . . . . . . . . 10 |- ( ph -> A. a e. ~P ( R1 ` A ) ( a =/= (/) -> ( y ` a ) e. ( ( ~P a i^i Fin ) \ { (/) } ) ) )
3		aomclem2.on	. . . . . . . . . . . . . 14 |- ( ph -> dom z e. On )
4		aomclem2.a	. . . . . . . . . . . . . 14 |- ( ph -> A e. On )
5	3, 4	jca 554	. . . . . . . . . . . . 13 |- ( ph -> ( dom z e. On /\ A e. On ) )
6		aomclem2.za	. . . . . . . . . . . . 13 |- ( ph -> dom z C_ A )
7		r1ord3 8645	. . . . . . . . . . . . 13 |- ( ( dom z e. On /\ A e. On ) -> ( dom z C_ A -> ( R1 ` dom z ) C_ ( R1 ` A ) ) )
8	5, 6, 7	sylc 65	. . . . . . . . . . . 12 |- ( ph -> ( R1 ` dom z ) C_ ( R1 ` A ) )
9		sspwb 4917	. . . . . . . . . . . 12 |- ( ( R1 ` dom z ) C_ ( R1 ` A ) <-> ~P ( R1 ` dom z ) C_ ~P ( R1 ` A ) )
10	8, 9	sylib 208	. . . . . . . . . . 11 |- ( ph -> ~P ( R1 ` dom z ) C_ ~P ( R1 ` A ) )
11	10	sseld 3602	. . . . . . . . . 10 |- ( ph -> ( a e. ~P ( R1 ` dom z ) -> a e. ~P ( R1 ` A ) ) )
12		rsp 2929	. . . . . . . . . 10 |- ( A. a e. ~P ( R1 ` A ) ( a =/= (/) -> ( y ` a ) e. ( ( ~P a i^i Fin ) \ { (/) } ) ) -> ( a e. ~P ( R1 ` A ) -> ( a =/= (/) -> ( y ` a ) e. ( ( ~P a i^i Fin ) \ { (/) } ) ) ) )
13	2, 11, 12	sylsyld 61	. . . . . . . . 9 |- ( ph -> ( a e. ~P ( R1 ` dom z ) -> ( a =/= (/) -> ( y ` a ) e. ( ( ~P a i^i Fin ) \ { (/) } ) ) ) )
14	13	3imp 1256	. . . . . . . 8 |- ( ( ph /\ a e. ~P ( R1 ` dom z ) /\ a =/= (/) ) -> ( y ` a ) e. ( ( ~P a i^i Fin ) \ { (/) } ) )
15	14	eldifad 3586	. . . . . . 7 |- ( ( ph /\ a e. ~P ( R1 ` dom z ) /\ a =/= (/) ) -> ( y ` a ) e. ( ~P a i^i Fin ) )
16		inss1 3833	. . . . . . . . 9 |- ( ~P a i^i Fin ) C_ ~P a
17	16	sseli 3599	. . . . . . . 8 |- ( ( y ` a ) e. ( ~P a i^i Fin ) -> ( y ` a ) e. ~P a )
18	17	elpwid 4170	. . . . . . 7 |- ( ( y ` a ) e. ( ~P a i^i Fin ) -> ( y ` a ) C_ a )
19	15, 18	syl 17	. . . . . 6 |- ( ( ph /\ a e. ~P ( R1 ` dom z ) /\ a =/= (/) ) -> ( y ` a ) C_ a )
20		aomclem2.b	. . . . . . . . 9 |- B = { <. a , b >. | E. c e. ( R1 ` U. dom z ) ( ( c e. b /\ -. c e. a ) /\ A. d e. ( R1 ` U. dom z ) ( d ( z ` U. dom z ) c -> ( d e. a <-> d e. b ) ) ) }
21		aomclem2.su	. . . . . . . . 9 |- ( ph -> dom z = suc U. dom z )
22		aomclem2.we	. . . . . . . . 9 |- ( ph -> A. a e. dom z ( z ` a ) We ( R1 ` a ) )
23	20, 3, 21, 22	aomclem1 37624	. . . . . . . 8 |- ( ph -> B Or ( R1 ` dom z ) )
24	23	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ a e. ~P ( R1 ` dom z ) /\ a =/= (/) ) -> B Or ( R1 ` dom z ) )
25		inss2 3834	. . . . . . . 8 |- ( ~P a i^i Fin ) C_ Fin
26	25, 15	sseldi 3601	. . . . . . 7 |- ( ( ph /\ a e. ~P ( R1 ` dom z ) /\ a =/= (/) ) -> ( y ` a ) e. Fin )
27		eldifsni 4320	. . . . . . . 8 |- ( ( y ` a ) e. ( ( ~P a i^i Fin ) \ { (/) } ) -> ( y ` a ) =/= (/) )
28	14, 27	syl 17	. . . . . . 7 |- ( ( ph /\ a e. ~P ( R1 ` dom z ) /\ a =/= (/) ) -> ( y ` a ) =/= (/) )
29		elpwi 4168	. . . . . . . . 9 |- ( a e. ~P ( R1 ` dom z ) -> a C_ ( R1 ` dom z ) )
30	29	3ad2ant2 1083	. . . . . . . 8 |- ( ( ph /\ a e. ~P ( R1 ` dom z ) /\ a =/= (/) ) -> a C_ ( R1 ` dom z ) )
31	19, 30	sstrd 3613	. . . . . . 7 |- ( ( ph /\ a e. ~P ( R1 ` dom z ) /\ a =/= (/) ) -> ( y ` a ) C_ ( R1 ` dom z ) )
32		fisupcl 8375	. . . . . . 7 |- ( ( B Or ( R1 ` dom z ) /\ ( ( y ` a ) e. Fin /\ ( y ` a ) =/= (/) /\ ( y ` a ) C_ ( R1 ` dom z ) ) ) -> sup ( ( y ` a ) , ( R1 ` dom z ) , B ) e. ( y ` a ) )
33	24, 26, 28, 31, 32	syl13anc 1328	. . . . . 6 |- ( ( ph /\ a e. ~P ( R1 ` dom z ) /\ a =/= (/) ) -> sup ( ( y ` a ) , ( R1 ` dom z ) , B ) e. ( y ` a ) )
34	19, 33	sseldd 3604	. . . . 5 |- ( ( ph /\ a e. ~P ( R1 ` dom z ) /\ a =/= (/) ) -> sup ( ( y ` a ) , ( R1 ` dom z ) , B ) e. a )
35		aomclem2.c	. . . . . 6 |- C = ( a e. _V |-> sup ( ( y ` a ) , ( R1 ` dom z ) , B ) )
36	35	fvmpt2 6291	. . . . 5 |- ( ( a e. _V /\ sup ( ( y ` a ) , ( R1 ` dom z ) , B ) e. a ) -> ( C ` a ) = sup ( ( y ` a ) , ( R1 ` dom z ) , B ) )
37	1, 34, 36	sylancr 695	. . . 4 |- ( ( ph /\ a e. ~P ( R1 ` dom z ) /\ a =/= (/) ) -> ( C ` a ) = sup ( ( y ` a ) , ( R1 ` dom z ) , B ) )
38	37, 34	eqeltrd 2701	. . 3 |- ( ( ph /\ a e. ~P ( R1 ` dom z ) /\ a =/= (/) ) -> ( C ` a ) e. a )
39	38	3exp 1264	. 2 |- ( ph -> ( a e. ~P ( R1 ` dom z ) -> ( a =/= (/) -> ( C ` a ) e. a ) ) )
40	39	ralrimiv 2965	1 |- ( ph -> A. a e. ~P ( R1 ` dom z ) ( a =/= (/) -> ( C ` a ) e. a ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   {copab 4712   |-> cmpt 4729   Or wor 5034   We wwe 5072   dom cdm 5114   Oncon0 5723   suc csuc 5725   ` cfv 5888   Fincfn 7955   supcsup 8346   R1cr1 8625

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-rmo 2920  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-sup 8348  df-r1 8627

This theorem is referenced by:  aomclem3  37626