Theorem aomclem3 37626

Description: Lemma for dfac11 37632. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Hypotheses

Ref	Expression
aomclem3.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 ) ) ) }
aomclem3.c	|- C = ( a e. _V |-> sup ( ( y ` a ) , ( R1 ` dom z ) , B ) )
aomclem3.d	|- D = recs ( ( a e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran a ) ) ) )
aomclem3.e	|- E = { <. a , b >. | |^| ( `' D " { a } ) e. |^| ( `' D " { b } ) }
aomclem3.on	|- ( ph -> dom z e. On )
aomclem3.su	|- ( ph -> dom z = suc U. dom z )
aomclem3.we	|- ( ph -> A. a e. dom z ( z ` a ) We ( R1 ` a ) )
aomclem3.a	|- ( ph -> A e. On )
aomclem3.za	|- ( ph -> dom z C_ A )
aomclem3.y	|- ( ph -> A. a e. ~P ( R1 ` A ) ( a =/= (/) -> ( y ` a ) e. ( ( ~P a i^i Fin ) \ { (/) } ) ) )

Assertion

Ref	Expression
aomclem3	|- ( ph -> E We ( R1 ` dom z ) )

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

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

Proof of Theorem aomclem3

Step	Hyp	Ref	Expression
1		aomclem3.d	. . 3 |- D = recs ( ( a e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran a ) ) ) )
2		rneq 5351	. . . . . . 7 |- ( a = c -> ran a = ran c )
3	2	difeq2d 3728	. . . . . 6 |- ( a = c -> ( ( R1 ` dom z ) \ ran a ) = ( ( R1 ` dom z ) \ ran c ) )
4	3	fveq2d 6195	. . . . 5 |- ( a = c -> ( C ` ( ( R1 ` dom z ) \ ran a ) ) = ( C ` ( ( R1 ` dom z ) \ ran c ) ) )
5	4	cbvmptv 4750	. . . 4 |- ( a e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran a ) ) ) = ( c e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran c ) ) )
6		recseq 7470	. . . 4 |- ( ( a e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran a ) ) ) = ( c e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran c ) ) ) -> recs ( ( a e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran a ) ) ) ) = recs ( ( c e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran c ) ) ) ) )
7	5, 6	ax-mp 5	. . 3 |- recs ( ( a e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran a ) ) ) ) = recs ( ( c e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran c ) ) ) )
8	1, 7	eqtri 2644	. 2 |- D = recs ( ( c e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran c ) ) ) )
9		fvexd 6203	. 2 |- ( ph -> ( R1 ` dom z ) e. _V )
10		aomclem3.b	. . . 4 |- 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 ) ) ) }
11		aomclem3.c	. . . 4 |- C = ( a e. _V |-> sup ( ( y ` a ) , ( R1 ` dom z ) , B ) )
12		aomclem3.on	. . . 4 |- ( ph -> dom z e. On )
13		aomclem3.su	. . . 4 |- ( ph -> dom z = suc U. dom z )
14		aomclem3.we	. . . 4 |- ( ph -> A. a e. dom z ( z ` a ) We ( R1 ` a ) )
15		aomclem3.a	. . . 4 |- ( ph -> A e. On )
16		aomclem3.za	. . . 4 |- ( ph -> dom z C_ A )
17		aomclem3.y	. . . 4 |- ( ph -> A. a e. ~P ( R1 ` A ) ( a =/= (/) -> ( y ` a ) e. ( ( ~P a i^i Fin ) \ { (/) } ) ) )
18	10, 11, 12, 13, 14, 15, 16, 17	aomclem2 37625	. . 3 |- ( ph -> A. a e. ~P ( R1 ` dom z ) ( a =/= (/) -> ( C ` a ) e. a ) )
19		neeq1 2856	. . . . 5 |- ( a = d -> ( a =/= (/) <-> d =/= (/) ) )
20		fveq2 6191	. . . . . 6 |- ( a = d -> ( C ` a ) = ( C ` d ) )
21		id 22	. . . . . 6 |- ( a = d -> a = d )
22	20, 21	eleq12d 2695	. . . . 5 |- ( a = d -> ( ( C ` a ) e. a <-> ( C ` d ) e. d ) )
23	19, 22	imbi12d 334	. . . 4 |- ( a = d -> ( ( a =/= (/) -> ( C ` a ) e. a ) <-> ( d =/= (/) -> ( C ` d ) e. d ) ) )
24	23	cbvralv 3171	. . 3 |- ( A. a e. ~P ( R1 ` dom z ) ( a =/= (/) -> ( C ` a ) e. a ) <-> A. d e. ~P ( R1 ` dom z ) ( d =/= (/) -> ( C ` d ) e. d ) )
25	18, 24	sylib 208	. 2 |- ( ph -> A. d e. ~P ( R1 ` dom z ) ( d =/= (/) -> ( C ` d ) e. d ) )
26		aomclem3.e	. 2 |- E = { <. a , b >. | |^| ( `' D " { a } ) e. |^| ( `' D " { b } ) }
27	8, 9, 25, 26	dnwech 37618	1 |- ( ph -> E We ( R1 ` dom z ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = 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   |^|cint 4475   class class class wbr 4653   {copab 4712   |-> cmpt 4729   We wwe 5072   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Oncon0 5723   suc csuc 5725   ` cfv 5888  recscrecs 7467   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-int 4476  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:  aomclem5  37628