Theorem cnfcom2lem 8598

Description: Lemma for cnfcom2 8599. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)

Hypotheses

Ref	Expression
cnfcom.s	|- S = dom ( om CNF A )
cnfcom.a	|- ( ph -> A e. On )
cnfcom.b	|- ( ph -> B e. ( om ^o A ) )
cnfcom.f	|- F = ( `' ( om CNF A ) ` B )
cnfcom.g	|- G = OrdIso ( _E , ( F supp (/) ) )
cnfcom.h	|- H = seq𝜔 ( ( k e. _V , z e. _V |-> ( M +o z ) ) , (/) )
cnfcom.t	|- T = seq𝜔 ( ( k e. _V , f e. _V |-> K ) , (/) )
cnfcom.m	|- M = ( ( om ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) )
cnfcom.k	|- K = ( ( x e. M |-> ( dom f +o x ) ) u. `' ( x e. dom f |-> ( M +o x ) ) )
cnfcom.w	|- W = ( G ` U. dom G )
cnfcom2.1	|- ( ph -> (/) e. B )

Assertion

Ref	Expression
cnfcom2lem	|- ( ph -> dom G = suc U. dom G )

Distinct variable groups:   x, k, z, A   x, M   f, k, x, z, F   z, T   x, W   f, G, k, x, z   f, H, x   S, k, z   ph, k, x, z

Allowed substitution hints:   ph( f)   A( f)   B( x, z, f, k)   S( x, f)   T( x, f, k)   H( z, k)   K( x, z, f, k)   M( z, f, k)   W( z, f, k)

Proof of Theorem cnfcom2lem

Step	Hyp	Ref	Expression
1		cnfcom2.1	. . . . . 6 |- ( ph -> (/) e. B )
2		n0i 3920	. . . . . 6 |- ( (/) e. B -> -. B = (/) )
3	1, 2	syl 17	. . . . 5 |- ( ph -> -. B = (/) )
4		cnfcom.f	. . . . . . . . . . . . . 14 |- F = ( `' ( om CNF A ) ` B )
5		cnfcom.s	. . . . . . . . . . . . . . . . 17 |- S = dom ( om CNF A )
6		omelon 8543	. . . . . . . . . . . . . . . . . 18 |- om e. On
7	6	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> om e. On )
8		cnfcom.a	. . . . . . . . . . . . . . . . 17 |- ( ph -> A e. On )
9	5, 7, 8	cantnff1o 8593	. . . . . . . . . . . . . . . 16 |- ( ph -> ( om CNF A ) : S -1-1-onto-> ( om ^o A ) )
10		f1ocnv 6149	. . . . . . . . . . . . . . . 16 |- ( ( om CNF A ) : S -1-1-onto-> ( om ^o A ) -> `' ( om CNF A ) : ( om ^o A ) -1-1-onto-> S )
11		f1of 6137	. . . . . . . . . . . . . . . 16 |- ( `' ( om CNF A ) : ( om ^o A ) -1-1-onto-> S -> `' ( om CNF A ) : ( om ^o A ) --> S )
12	9, 10, 11	3syl 18	. . . . . . . . . . . . . . 15 |- ( ph -> `' ( om CNF A ) : ( om ^o A ) --> S )
13		cnfcom.b	. . . . . . . . . . . . . . 15 |- ( ph -> B e. ( om ^o A ) )
14	12, 13	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ph -> ( `' ( om CNF A ) ` B ) e. S )
15	4, 14	syl5eqel 2705	. . . . . . . . . . . . 13 |- ( ph -> F e. S )
16	5, 7, 8	cantnfs 8563	. . . . . . . . . . . . 13 |- ( ph -> ( F e. S <-> ( F : A --> om /\ F finSupp (/) ) ) )
17	15, 16	mpbid 222	. . . . . . . . . . . 12 |- ( ph -> ( F : A --> om /\ F finSupp (/) ) )
18	17	simpld 475	. . . . . . . . . . 11 |- ( ph -> F : A --> om )
19	18	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ dom G = (/) ) -> F : A --> om )
20	19	feqmptd 6249	. . . . . . . . 9 |- ( ( ph /\ dom G = (/) ) -> F = ( x e. A |-> ( F ` x ) ) )
21		dif0 3950	. . . . . . . . . . . 12 |- ( A \ (/) ) = A
22	21	eleq2i 2693	. . . . . . . . . . 11 |- ( x e. ( A \ (/) ) <-> x e. A )
23		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ dom G = (/) ) -> dom G = (/) )
24		suppssdm 7308	. . . . . . . . . . . . . . . . . . . 20 |- ( F supp (/) ) C_ dom F
25		fdm 6051	. . . . . . . . . . . . . . . . . . . . 21 |- ( F : A --> om -> dom F = A )
26	18, 25	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> dom F = A )
27	24, 26	syl5sseq 3653	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( F supp (/) ) C_ A )
28	8, 27	ssexd 4805	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( F supp (/) ) e. _V )
29		cnfcom.g	. . . . . . . . . . . . . . . . . . . 20 |- G = OrdIso ( _E , ( F supp (/) ) )
30	5, 7, 8, 29, 15	cantnfcl 8564	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( _E We ( F supp (/) ) /\ dom G e. om ) )
31	30	simpld 475	. . . . . . . . . . . . . . . . . 18 |- ( ph -> _E We ( F supp (/) ) )
32	29	oien 8443	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F supp (/) ) e. _V /\ _E We ( F supp (/) ) ) -> dom G ~~ ( F supp (/) ) )
33	28, 31, 32	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ph -> dom G ~~ ( F supp (/) ) )
34	33	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ dom G = (/) ) -> dom G ~~ ( F supp (/) ) )
35	23, 34	eqbrtrrd 4677	. . . . . . . . . . . . . . 15 |- ( ( ph /\ dom G = (/) ) -> (/) ~~ ( F supp (/) ) )
36	35	ensymd 8007	. . . . . . . . . . . . . 14 |- ( ( ph /\ dom G = (/) ) -> ( F supp (/) ) ~~ (/) )
37		en0 8019	. . . . . . . . . . . . . 14 |- ( ( F supp (/) ) ~~ (/) <-> ( F supp (/) ) = (/) )
38	36, 37	sylib 208	. . . . . . . . . . . . 13 |- ( ( ph /\ dom G = (/) ) -> ( F supp (/) ) = (/) )
39		ss0b 3973	. . . . . . . . . . . . 13 |- ( ( F supp (/) ) C_ (/) <-> ( F supp (/) ) = (/) )
40	38, 39	sylibr 224	. . . . . . . . . . . 12 |- ( ( ph /\ dom G = (/) ) -> ( F supp (/) ) C_ (/) )
41	8	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ dom G = (/) ) -> A e. On )
42		0ex 4790	. . . . . . . . . . . . 13 |- (/) e. _V
43	42	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ dom G = (/) ) -> (/) e. _V )
44	19, 40, 41, 43	suppssr 7326	. . . . . . . . . . 11 |- ( ( ( ph /\ dom G = (/) ) /\ x e. ( A \ (/) ) ) -> ( F ` x ) = (/) )
45	22, 44	sylan2br 493	. . . . . . . . . 10 |- ( ( ( ph /\ dom G = (/) ) /\ x e. A ) -> ( F ` x ) = (/) )
46	45	mpteq2dva 4744	. . . . . . . . 9 |- ( ( ph /\ dom G = (/) ) -> ( x e. A |-> ( F ` x ) ) = ( x e. A |-> (/) ) )
47	20, 46	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ dom G = (/) ) -> F = ( x e. A |-> (/) ) )
48		fconstmpt 5163	. . . . . . . 8 |- ( A X. { (/) } ) = ( x e. A |-> (/) )
49	47, 48	syl6eqr 2674	. . . . . . 7 |- ( ( ph /\ dom G = (/) ) -> F = ( A X. { (/) } ) )
50	49	fveq2d 6195	. . . . . 6 |- ( ( ph /\ dom G = (/) ) -> ( ( om CNF A ) ` F ) = ( ( om CNF A ) ` ( A X. { (/) } ) ) )
51	4	fveq2i 6194	. . . . . . . 8 |- ( ( om CNF A ) ` F ) = ( ( om CNF A ) ` ( `' ( om CNF A ) ` B ) )
52		f1ocnvfv2 6533	. . . . . . . . 9 |- ( ( ( om CNF A ) : S -1-1-onto-> ( om ^o A ) /\ B e. ( om ^o A ) ) -> ( ( om CNF A ) ` ( `' ( om CNF A ) ` B ) ) = B )
53	9, 13, 52	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( om CNF A ) ` ( `' ( om CNF A ) ` B ) ) = B )
54	51, 53	syl5eq 2668	. . . . . . 7 |- ( ph -> ( ( om CNF A ) ` F ) = B )
55	54	adantr 481	. . . . . 6 |- ( ( ph /\ dom G = (/) ) -> ( ( om CNF A ) ` F ) = B )
56		peano1 7085	. . . . . . . . 9 |- (/) e. om
57	56	a1i 11	. . . . . . . 8 |- ( ph -> (/) e. om )
58	5, 7, 8, 57	cantnf0 8572	. . . . . . 7 |- ( ph -> ( ( om CNF A ) ` ( A X. { (/) } ) ) = (/) )
59	58	adantr 481	. . . . . 6 |- ( ( ph /\ dom G = (/) ) -> ( ( om CNF A ) ` ( A X. { (/) } ) ) = (/) )
60	50, 55, 59	3eqtr3d 2664	. . . . 5 |- ( ( ph /\ dom G = (/) ) -> B = (/) )
61	3, 60	mtand 691	. . . 4 |- ( ph -> -. dom G = (/) )
62	30	simprd 479	. . . . 5 |- ( ph -> dom G e. om )
63		nnlim 7078	. . . . 5 |- ( dom G e. om -> -. Lim dom G )
64	62, 63	syl 17	. . . 4 |- ( ph -> -. Lim dom G )
65		ioran 511	. . . 4 |- ( -. ( dom G = (/) \/ Lim dom G ) <-> ( -. dom G = (/) /\ -. Lim dom G ) )
66	61, 64, 65	sylanbrc 698	. . 3 |- ( ph -> -. ( dom G = (/) \/ Lim dom G ) )
67	29	oicl 8434	. . . 4 |- Ord dom G
68		unizlim 5844	. . . 4 |- ( Ord dom G -> ( dom G = U. dom G <-> ( dom G = (/) \/ Lim dom G ) ) )
69	67, 68	ax-mp 5	. . 3 |- ( dom G = U. dom G <-> ( dom G = (/) \/ Lim dom G ) )
70	66, 69	sylnibr 319	. 2 |- ( ph -> -. dom G = U. dom G )
71		orduniorsuc 7030	. . . 4 |- ( Ord dom G -> ( dom G = U. dom G \/ dom G = suc U. dom G ) )
72	67, 71	mp1i 13	. . 3 |- ( ph -> ( dom G = U. dom G \/ dom G = suc U. dom G ) )
73	72	ord 392	. 2 |- ( ph -> ( -. dom G = U. dom G -> dom G = suc U. dom G ) )
74	70, 73	mpd 15	1 |- ( ph -> dom G = suc U. dom G )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   _E cep 5028   We wwe 5072   X. cxp 5112   `'ccnv 5113   dom cdm 5114   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   supp csupp 7295  seq_𝜔cseqom 7542   +o coa 7557   .o comu 7558   ^o coe 7559   ~~ cen 7952   finSupp cfsupp 8275  OrdIsocoi 8414   CNF ccnf 8558

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  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-se 5074  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-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-cnf 8559

This theorem is referenced by:  cnfcom2  8599  cnfcom3lem  8600  cnfcom3  8601