Theorem aomclem4 37627

Description: Lemma for dfac11 37632. Limit case. Patch together well-orderings constructed so far using fnwe2 37623 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Hypotheses

Ref	Expression
aomclem4.f	|- F = { <. a , b >. | ( ( rank ` a ) _E ( rank ` b ) \/ ( ( rank ` a ) = ( rank ` b ) /\ a ( z ` suc ( rank ` a ) ) b ) ) }
aomclem4.on	|- ( ph -> dom z e. On )
aomclem4.su	|- ( ph -> dom z = U. dom z )
aomclem4.we	|- ( ph -> A. a e. dom z ( z ` a ) We ( R1 ` a ) )

Assertion

Ref	Expression
aomclem4	|- ( ph -> F We ( R1 ` dom z ) )

Distinct variable groups:   z, a, b   ph, a, b

Allowed substitution hints:   ph( z)   F( z, a, b)

Proof of Theorem aomclem4

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suceq 5790	. . 3 |- ( c = ( rank ` a ) -> suc c = suc ( rank ` a ) )
2	1	fveq2d 6195	. 2 |- ( c = ( rank ` a ) -> ( z ` suc c ) = ( z ` suc ( rank ` a ) ) )
3		aomclem4.f	. 2 |- F = { <. a , b >. | ( ( rank ` a ) _E ( rank ` b ) \/ ( ( rank ` a ) = ( rank ` b ) /\ a ( z ` suc ( rank ` a ) ) b ) ) }
4		r1fnon 8630	. . . . . . . . . . . . . 14 |- R1 Fn On
5		fnfun 5988	. . . . . . . . . . . . . 14 |- ( R1 Fn On -> Fun R1 )
6	4, 5	ax-mp 5	. . . . . . . . . . . . 13 |- Fun R1
7		fndm 5990	. . . . . . . . . . . . . . 15 |- ( R1 Fn On -> dom R1 = On )
8	4, 7	ax-mp 5	. . . . . . . . . . . . . 14 |- dom R1 = On
9	8	eqimss2i 3660	. . . . . . . . . . . . 13 |- On C_ dom R1
10	6, 9	pm3.2i 471	. . . . . . . . . . . 12 |- ( Fun R1 /\ On C_ dom R1 )
11		aomclem4.on	. . . . . . . . . . . 12 |- ( ph -> dom z e. On )
12		funfvima2 6493	. . . . . . . . . . . 12 |- ( ( Fun R1 /\ On C_ dom R1 ) -> ( dom z e. On -> ( R1 ` dom z ) e. ( R1 " On ) ) )
13	10, 11, 12	mpsyl 68	. . . . . . . . . . 11 |- ( ph -> ( R1 ` dom z ) e. ( R1 " On ) )
14		elssuni 4467	. . . . . . . . . . 11 |- ( ( R1 ` dom z ) e. ( R1 " On ) -> ( R1 ` dom z ) C_ U. ( R1 " On ) )
15	13, 14	syl 17	. . . . . . . . . 10 |- ( ph -> ( R1 ` dom z ) C_ U. ( R1 " On ) )
16	15	sselda 3603	. . . . . . . . 9 |- ( ( ph /\ b e. ( R1 ` dom z ) ) -> b e. U. ( R1 " On ) )
17		rankidb 8663	. . . . . . . . 9 |- ( b e. U. ( R1 " On ) -> b e. ( R1 ` suc ( rank ` b ) ) )
18	16, 17	syl 17	. . . . . . . 8 |- ( ( ph /\ b e. ( R1 ` dom z ) ) -> b e. ( R1 ` suc ( rank ` b ) ) )
19		suceq 5790	. . . . . . . . . 10 |- ( ( rank ` b ) = ( rank ` a ) -> suc ( rank ` b ) = suc ( rank ` a ) )
20	19	fveq2d 6195	. . . . . . . . 9 |- ( ( rank ` b ) = ( rank ` a ) -> ( R1 ` suc ( rank ` b ) ) = ( R1 ` suc ( rank ` a ) ) )
21	20	eleq2d 2687	. . . . . . . 8 |- ( ( rank ` b ) = ( rank ` a ) -> ( b e. ( R1 ` suc ( rank ` b ) ) <-> b e. ( R1 ` suc ( rank ` a ) ) ) )
22	18, 21	syl5ibcom 235	. . . . . . 7 |- ( ( ph /\ b e. ( R1 ` dom z ) ) -> ( ( rank ` b ) = ( rank ` a ) -> b e. ( R1 ` suc ( rank ` a ) ) ) )
23	22	expimpd 629	. . . . . 6 |- ( ph -> ( ( b e. ( R1 ` dom z ) /\ ( rank ` b ) = ( rank ` a ) ) -> b e. ( R1 ` suc ( rank ` a ) ) ) )
24	23	ss2abdv 3675	. . . . 5 |- ( ph -> { b | ( b e. ( R1 ` dom z ) /\ ( rank ` b ) = ( rank ` a ) ) } C_ { b | b e. ( R1 ` suc ( rank ` a ) ) } )
25		df-rab 2921	. . . . 5 |- { b e. ( R1 ` dom z ) | ( rank ` b ) = ( rank ` a ) } = { b | ( b e. ( R1 ` dom z ) /\ ( rank ` b ) = ( rank ` a ) ) }
26		abid1 2744	. . . . 5 |- ( R1 ` suc ( rank ` a ) ) = { b | b e. ( R1 ` suc ( rank ` a ) ) }
27	24, 25, 26	3sstr4g 3646	. . . 4 |- ( ph -> { b e. ( R1 ` dom z ) | ( rank ` b ) = ( rank ` a ) } C_ ( R1 ` suc ( rank ` a ) ) )
28	27	adantr 481	. . 3 |- ( ( ph /\ a e. ( R1 ` dom z ) ) -> { b e. ( R1 ` dom z ) | ( rank ` b ) = ( rank ` a ) } C_ ( R1 ` suc ( rank ` a ) ) )
29		rankr1ai 8661	. . . . . 6 |- ( a e. ( R1 ` dom z ) -> ( rank ` a ) e. dom z )
30	29	adantl 482	. . . . 5 |- ( ( ph /\ a e. ( R1 ` dom z ) ) -> ( rank ` a ) e. dom z )
31		eloni 5733	. . . . . . . 8 |- ( dom z e. On -> Ord dom z )
32	11, 31	syl 17	. . . . . . 7 |- ( ph -> Ord dom z )
33		aomclem4.su	. . . . . . 7 |- ( ph -> dom z = U. dom z )
34		limsuc2 37611	. . . . . . 7 |- ( ( Ord dom z /\ dom z = U. dom z ) -> ( ( rank ` a ) e. dom z <-> suc ( rank ` a ) e. dom z ) )
35	32, 33, 34	syl2anc 693	. . . . . 6 |- ( ph -> ( ( rank ` a ) e. dom z <-> suc ( rank ` a ) e. dom z ) )
36	35	adantr 481	. . . . 5 |- ( ( ph /\ a e. ( R1 ` dom z ) ) -> ( ( rank ` a ) e. dom z <-> suc ( rank ` a ) e. dom z ) )
37	30, 36	mpbid 222	. . . 4 |- ( ( ph /\ a e. ( R1 ` dom z ) ) -> suc ( rank ` a ) e. dom z )
38		aomclem4.we	. . . . . 6 |- ( ph -> A. a e. dom z ( z ` a ) We ( R1 ` a ) )
39		fveq2 6191	. . . . . . . 8 |- ( a = b -> ( z ` a ) = ( z ` b ) )
40		fveq2 6191	. . . . . . . 8 |- ( a = b -> ( R1 ` a ) = ( R1 ` b ) )
41	39, 40	weeq12d 37610	. . . . . . 7 |- ( a = b -> ( ( z ` a ) We ( R1 ` a ) <-> ( z ` b ) We ( R1 ` b ) ) )
42	41	cbvralv 3171	. . . . . 6 |- ( A. a e. dom z ( z ` a ) We ( R1 ` a ) <-> A. b e. dom z ( z ` b ) We ( R1 ` b ) )
43	38, 42	sylib 208	. . . . 5 |- ( ph -> A. b e. dom z ( z ` b ) We ( R1 ` b ) )
44	43	adantr 481	. . . 4 |- ( ( ph /\ a e. ( R1 ` dom z ) ) -> A. b e. dom z ( z ` b ) We ( R1 ` b ) )
45		fveq2 6191	. . . . . 6 |- ( b = suc ( rank ` a ) -> ( z ` b ) = ( z ` suc ( rank ` a ) ) )
46		fveq2 6191	. . . . . 6 |- ( b = suc ( rank ` a ) -> ( R1 ` b ) = ( R1 ` suc ( rank ` a ) ) )
47	45, 46	weeq12d 37610	. . . . 5 |- ( b = suc ( rank ` a ) -> ( ( z ` b ) We ( R1 ` b ) <-> ( z ` suc ( rank ` a ) ) We ( R1 ` suc ( rank ` a ) ) ) )
48	47	rspcva 3307	. . . 4 |- ( ( suc ( rank ` a ) e. dom z /\ A. b e. dom z ( z ` b ) We ( R1 ` b ) ) -> ( z ` suc ( rank ` a ) ) We ( R1 ` suc ( rank ` a ) ) )
49	37, 44, 48	syl2anc 693	. . 3 |- ( ( ph /\ a e. ( R1 ` dom z ) ) -> ( z ` suc ( rank ` a ) ) We ( R1 ` suc ( rank ` a ) ) )
50		wess 5101	. . 3 |- ( { b e. ( R1 ` dom z ) | ( rank ` b ) = ( rank ` a ) } C_ ( R1 ` suc ( rank ` a ) ) -> ( ( z ` suc ( rank ` a ) ) We ( R1 ` suc ( rank ` a ) ) -> ( z ` suc ( rank ` a ) ) We { b e. ( R1 ` dom z ) | ( rank ` b ) = ( rank ` a ) } ) )
51	28, 49, 50	sylc 65	. 2 |- ( ( ph /\ a e. ( R1 ` dom z ) ) -> ( z ` suc ( rank ` a ) ) We { b e. ( R1 ` dom z ) | ( rank ` b ) = ( rank ` a ) } )
52		rankf 8657	. . . 4 |- rank : U. ( R1 " On ) --> On
53	52	a1i 11	. . 3 |- ( ph -> rank : U. ( R1 " On ) --> On )
54	53, 15	fssresd 6071	. 2 |- ( ph -> ( rank |` ( R1 ` dom z ) ) : ( R1 ` dom z ) --> On )
55		epweon 6983	. . 3 |- _E We On
56	55	a1i 11	. 2 |- ( ph -> _E We On )
57	2, 3, 51, 54, 56	fnwe2 37623	1 |- ( ph -> F We ( R1 ` dom z ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   C_ wss 3574   U.cuni 4436   class class class wbr 4653   {copab 4712   _E cep 5028   We wwe 5072   dom cdm 5114   "cima 5117   Ord word 5722   Oncon0 5723   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   R1cr1 8625   rankcrnk 8626

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-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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  aomclem5  37628