Theorem cantnfcl 8564

Description: Basic properties of the order isomorphism G used later. The support of an F e. S is a finite subset of A, so it is well-ordered by _E and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)

Hypotheses

Ref	Expression
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )
cantnfcl.g	|- G = OrdIso ( _E , ( F supp (/) ) )
cantnfcl.f	|- ( ph -> F e. S )

Assertion

Ref	Expression
cantnfcl	|- ( ph -> ( _E We ( F supp (/) ) /\ dom G e. om ) )

Proof of Theorem cantnfcl

Step	Hyp	Ref	Expression
1		suppssdm 7308	. . . . 5 |- ( F supp (/) ) C_ dom F
2		cantnfcl.f	. . . . . . . 8 |- ( ph -> F e. S )
3		cantnfs.s	. . . . . . . . 9 |- S = dom ( A CNF B )
4		cantnfs.a	. . . . . . . . 9 |- ( ph -> A e. On )
5		cantnfs.b	. . . . . . . . 9 |- ( ph -> B e. On )
6	3, 4, 5	cantnfs 8563	. . . . . . . 8 |- ( ph -> ( F e. S <-> ( F : B --> A /\ F finSupp (/) ) ) )
7	2, 6	mpbid 222	. . . . . . 7 |- ( ph -> ( F : B --> A /\ F finSupp (/) ) )
8	7	simpld 475	. . . . . 6 |- ( ph -> F : B --> A )
9		fdm 6051	. . . . . 6 |- ( F : B --> A -> dom F = B )
10	8, 9	syl 17	. . . . 5 |- ( ph -> dom F = B )
11	1, 10	syl5sseq 3653	. . . 4 |- ( ph -> ( F supp (/) ) C_ B )
12		onss 6990	. . . . 5 |- ( B e. On -> B C_ On )
13	5, 12	syl 17	. . . 4 |- ( ph -> B C_ On )
14	11, 13	sstrd 3613	. . 3 |- ( ph -> ( F supp (/) ) C_ On )
15		epweon 6983	. . 3 |- _E We On
16		wess 5101	. . 3 |- ( ( F supp (/) ) C_ On -> ( _E We On -> _E We ( F supp (/) ) ) )
17	14, 15, 16	mpisyl 21	. 2 |- ( ph -> _E We ( F supp (/) ) )
18		ovexd 6680	. . . . 5 |- ( ph -> ( F supp (/) ) e. _V )
19		cantnfcl.g	. . . . . 6 |- G = OrdIso ( _E , ( F supp (/) ) )
20	19	oion 8441	. . . . 5 |- ( ( F supp (/) ) e. _V -> dom G e. On )
21	18, 20	syl 17	. . . 4 |- ( ph -> dom G e. On )
22	7	simprd 479	. . . . . 6 |- ( ph -> F finSupp (/) )
23	22	fsuppimpd 8282	. . . . 5 |- ( ph -> ( F supp (/) ) e. Fin )
24	19	oien 8443	. . . . . 6 |- ( ( ( F supp (/) ) e. _V /\ _E We ( F supp (/) ) ) -> dom G ~~ ( F supp (/) ) )
25	18, 17, 24	syl2anc 693	. . . . 5 |- ( ph -> dom G ~~ ( F supp (/) ) )
26		enfii 8177	. . . . 5 |- ( ( ( F supp (/) ) e. Fin /\ dom G ~~ ( F supp (/) ) ) -> dom G e. Fin )
27	23, 25, 26	syl2anc 693	. . . 4 |- ( ph -> dom G e. Fin )
28	21, 27	elind 3798	. . 3 |- ( ph -> dom G e. ( On i^i Fin ) )
29		onfin2 8152	. . 3 |- om = ( On i^i Fin )
30	28, 29	syl6eleqr 2712	. 2 |- ( ph -> dom G e. om )
31	17, 30	jca 554	1 |- ( ph -> ( _E We ( F supp (/) ) /\ dom G e. om ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   _E cep 5028   We wwe 5072   dom cdm 5114   Oncon0 5723   -->wf 5884  (class class class)co 6650   omcom 7065   supp csupp 7295   ~~ cen 7952   Fincfn 7955   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-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-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:  cantnfval2  8566  cantnfle  8568  cantnflt  8569  cantnflt2  8570  cantnff  8571  cantnfp1lem2  8576  cantnfp1lem3  8577  cantnflem1b  8583  cantnflem1d  8585  cantnflem1  8586  cnfcomlem  8596  cnfcom  8597  cnfcom2lem  8598  cnfcom3lem  8600