Theorem cantnfp1lem2 8576

Description: Lemma for cantnfp1 8578. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 30-Jun-2019.)

Hypotheses

Ref	Expression
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )
cantnfp1.g	|- ( ph -> G e. S )
cantnfp1.x	|- ( ph -> X e. B )
cantnfp1.y	|- ( ph -> Y e. A )
cantnfp1.s	|- ( ph -> ( G supp (/) ) C_ X )
cantnfp1.f	|- F = ( t e. B |-> if ( t = X , Y , ( G ` t ) ) )
cantnfp1.e	|- ( ph -> (/) e. Y )
cantnfp1.o	|- O = OrdIso ( _E , ( F supp (/) ) )

Assertion

Ref	Expression
cantnfp1lem2	|- ( ph -> dom O = suc U. dom O )

Distinct variable groups:   t, B   t, A   t, S   t, G   ph, t   t, Y   t, X

Allowed substitution hints:   F( t)   O( t)

Proof of Theorem cantnfp1lem2

Step	Hyp	Ref	Expression
1		cantnfp1.x	. . . . . . 7 |- ( ph -> X e. B )
2		cantnfp1.y	. . . . . . . . 9 |- ( ph -> Y e. A )
3		iftrue 4092	. . . . . . . . . 10 |- ( t = X -> if ( t = X , Y , ( G ` t ) ) = Y )
4		cantnfp1.f	. . . . . . . . . 10 |- F = ( t e. B |-> if ( t = X , Y , ( G ` t ) ) )
5	3, 4	fvmptg 6280	. . . . . . . . 9 |- ( ( X e. B /\ Y e. A ) -> ( F ` X ) = Y )
6	1, 2, 5	syl2anc 693	. . . . . . . 8 |- ( ph -> ( F ` X ) = Y )
7		cantnfp1.e	. . . . . . . . 9 |- ( ph -> (/) e. Y )
8		cantnfs.a	. . . . . . . . . . 11 |- ( ph -> A e. On )
9		onelon 5748	. . . . . . . . . . 11 |- ( ( A e. On /\ Y e. A ) -> Y e. On )
10	8, 2, 9	syl2anc 693	. . . . . . . . . 10 |- ( ph -> Y e. On )
11		on0eln0 5780	. . . . . . . . . 10 |- ( Y e. On -> ( (/) e. Y <-> Y =/= (/) ) )
12	10, 11	syl 17	. . . . . . . . 9 |- ( ph -> ( (/) e. Y <-> Y =/= (/) ) )
13	7, 12	mpbid 222	. . . . . . . 8 |- ( ph -> Y =/= (/) )
14	6, 13	eqnetrd 2861	. . . . . . 7 |- ( ph -> ( F ` X ) =/= (/) )
15	2	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ t e. B ) -> Y e. A )
16		cantnfp1.g	. . . . . . . . . . . . . 14 |- ( ph -> G e. S )
17		cantnfs.s	. . . . . . . . . . . . . . 15 |- S = dom ( A CNF B )
18		cantnfs.b	. . . . . . . . . . . . . . 15 |- ( ph -> B e. On )
19	17, 8, 18	cantnfs 8563	. . . . . . . . . . . . . 14 |- ( ph -> ( G e. S <-> ( G : B --> A /\ G finSupp (/) ) ) )
20	16, 19	mpbid 222	. . . . . . . . . . . . 13 |- ( ph -> ( G : B --> A /\ G finSupp (/) ) )
21	20	simpld 475	. . . . . . . . . . . 12 |- ( ph -> G : B --> A )
22	21	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ t e. B ) -> ( G ` t ) e. A )
23	15, 22	ifcld 4131	. . . . . . . . . 10 |- ( ( ph /\ t e. B ) -> if ( t = X , Y , ( G ` t ) ) e. A )
24	23, 4	fmptd 6385	. . . . . . . . 9 |- ( ph -> F : B --> A )
25		ffn 6045	. . . . . . . . 9 |- ( F : B --> A -> F Fn B )
26	24, 25	syl 17	. . . . . . . 8 |- ( ph -> F Fn B )
27		0ex 4790	. . . . . . . . 9 |- (/) e. _V
28	27	a1i 11	. . . . . . . 8 |- ( ph -> (/) e. _V )
29		elsuppfn 7303	. . . . . . . 8 |- ( ( F Fn B /\ B e. On /\ (/) e. _V ) -> ( X e. ( F supp (/) ) <-> ( X e. B /\ ( F ` X ) =/= (/) ) ) )
30	26, 18, 28, 29	syl3anc 1326	. . . . . . 7 |- ( ph -> ( X e. ( F supp (/) ) <-> ( X e. B /\ ( F ` X ) =/= (/) ) ) )
31	1, 14, 30	mpbir2and 957	. . . . . 6 |- ( ph -> X e. ( F supp (/) ) )
32		n0i 3920	. . . . . 6 |- ( X e. ( F supp (/) ) -> -. ( F supp (/) ) = (/) )
33	31, 32	syl 17	. . . . 5 |- ( ph -> -. ( F supp (/) ) = (/) )
34		suppssdm 7308	. . . . . . . . 9 |- ( F supp (/) ) C_ dom F
35	4, 23	dmmptd 6024	. . . . . . . . 9 |- ( ph -> dom F = B )
36	34, 35	syl5sseq 3653	. . . . . . . 8 |- ( ph -> ( F supp (/) ) C_ B )
37	18, 36	ssexd 4805	. . . . . . 7 |- ( ph -> ( F supp (/) ) e. _V )
38		cantnfp1.o	. . . . . . . . 9 |- O = OrdIso ( _E , ( F supp (/) ) )
39		cantnfp1.s	. . . . . . . . . 10 |- ( ph -> ( G supp (/) ) C_ X )
40	17, 8, 18, 16, 1, 2, 39, 4	cantnfp1lem1 8575	. . . . . . . . 9 |- ( ph -> F e. S )
41	17, 8, 18, 38, 40	cantnfcl 8564	. . . . . . . 8 |- ( ph -> ( _E We ( F supp (/) ) /\ dom O e. om ) )
42	41	simpld 475	. . . . . . 7 |- ( ph -> _E We ( F supp (/) ) )
43	38	oien 8443	. . . . . . 7 |- ( ( ( F supp (/) ) e. _V /\ _E We ( F supp (/) ) ) -> dom O ~~ ( F supp (/) ) )
44	37, 42, 43	syl2anc 693	. . . . . 6 |- ( ph -> dom O ~~ ( F supp (/) ) )
45		breq1 4656	. . . . . . 7 |- ( dom O = (/) -> ( dom O ~~ ( F supp (/) ) <-> (/) ~~ ( F supp (/) ) ) )
46		ensymb 8004	. . . . . . . 8 |- ( (/) ~~ ( F supp (/) ) <-> ( F supp (/) ) ~~ (/) )
47		en0 8019	. . . . . . . 8 |- ( ( F supp (/) ) ~~ (/) <-> ( F supp (/) ) = (/) )
48	46, 47	bitri 264	. . . . . . 7 |- ( (/) ~~ ( F supp (/) ) <-> ( F supp (/) ) = (/) )
49	45, 48	syl6bb 276	. . . . . 6 |- ( dom O = (/) -> ( dom O ~~ ( F supp (/) ) <-> ( F supp (/) ) = (/) ) )
50	44, 49	syl5ibcom 235	. . . . 5 |- ( ph -> ( dom O = (/) -> ( F supp (/) ) = (/) ) )
51	33, 50	mtod 189	. . . 4 |- ( ph -> -. dom O = (/) )
52	41	simprd 479	. . . . 5 |- ( ph -> dom O e. om )
53		nnlim 7078	. . . . 5 |- ( dom O e. om -> -. Lim dom O )
54	52, 53	syl 17	. . . 4 |- ( ph -> -. Lim dom O )
55		ioran 511	. . . 4 |- ( -. ( dom O = (/) \/ Lim dom O ) <-> ( -. dom O = (/) /\ -. Lim dom O ) )
56	51, 54, 55	sylanbrc 698	. . 3 |- ( ph -> -. ( dom O = (/) \/ Lim dom O ) )
57		nnord 7073	. . . 4 |- ( dom O e. om -> Ord dom O )
58		unizlim 5844	. . . 4 |- ( Ord dom O -> ( dom O = U. dom O <-> ( dom O = (/) \/ Lim dom O ) ) )
59	52, 57, 58	3syl 18	. . 3 |- ( ph -> ( dom O = U. dom O <-> ( dom O = (/) \/ Lim dom O ) ) )
60	56, 59	mtbird 315	. 2 |- ( ph -> -. dom O = U. dom O )
61		orduniorsuc 7030	. . . 4 |- ( Ord dom O -> ( dom O = U. dom O \/ dom O = suc U. dom O ) )
62	52, 57, 61	3syl 18	. . 3 |- ( ph -> ( dom O = U. dom O \/ dom O = suc U. dom O ) )
63	62	ord 392	. 2 |- ( ph -> ( -. dom O = U. dom O -> dom O = suc U. dom O ) )
64	60, 63	mpd 15	1 |- ( ph -> dom O = suc U. dom O )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ifcif 4086   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   _E cep 5028   We wwe 5072   dom cdm 5114   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   supp csupp 7295   ~~ 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

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-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-oadd 7564  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:  cantnfp1lem3  8577