Theorem cantnflt2 8570

Description: An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)

Hypotheses

Ref	Expression
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )
cantnflt2.f	|- ( ph -> F e. S )
cantnflt2.a	|- ( ph -> (/) e. A )
cantnflt2.c	|- ( ph -> C e. On )
cantnflt2.s	|- ( ph -> ( F supp (/) ) C_ C )

Assertion

Ref	Expression
cantnflt2	|- ( ph -> ( ( A CNF B ) ` F ) e. ( A ^o C ) )

Proof of Theorem cantnflt2

Dummy variables k z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfs.s	. . 3 |- S = dom ( A CNF B )
2		cantnfs.a	. . 3 |- ( ph -> A e. On )
3		cantnfs.b	. . 3 |- ( ph -> B e. On )
4		eqid 2622	. . 3 |- OrdIso ( _E , ( F supp (/) ) ) = OrdIso ( _E , ( F supp (/) ) )
5		cantnflt2.f	. . 3 |- ( ph -> F e. S )
6		eqid 2622	. . 3 |- seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( F supp (/) ) ) ` k ) ) .o ( F ` (OrdIso ( _E , ( F supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) = seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( F supp (/) ) ) ` k ) ) .o ( F ` (OrdIso ( _E , ( F supp (/) ) ) ` k ) ) ) +o z ) ) , (/) )
7	1, 2, 3, 4, 5, 6	cantnfval 8565	. 2 |- ( ph -> ( ( A CNF B ) ` F ) = (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( F supp (/) ) ) ` k ) ) .o ( F ` (OrdIso ( _E , ( F supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` dom OrdIso ( _E , ( F supp (/) ) ) ) )
8		cantnflt2.a	. . 3 |- ( ph -> (/) e. A )
9		cantnflt2.c	. . . . 5 |- ( ph -> C e. On )
10		cantnflt2.s	. . . . 5 |- ( ph -> ( F supp (/) ) C_ C )
11	9, 10	ssexd 4805	. . . 4 |- ( ph -> ( F supp (/) ) e. _V )
12	4	oion 8441	. . . 4 |- ( ( F supp (/) ) e. _V -> dom OrdIso ( _E , ( F supp (/) ) ) e. On )
13		sucidg 5803	. . . 4 |- ( dom OrdIso ( _E , ( F supp (/) ) ) e. On -> dom OrdIso ( _E , ( F supp (/) ) ) e. suc dom OrdIso ( _E , ( F supp (/) ) ) )
14	11, 12, 13	3syl 18	. . 3 |- ( ph -> dom OrdIso ( _E , ( F supp (/) ) ) e. suc dom OrdIso ( _E , ( F supp (/) ) ) )
15	1, 2, 3, 4, 5	cantnfcl 8564	. . . . . . 7 |- ( ph -> ( _E We ( F supp (/) ) /\ dom OrdIso ( _E , ( F supp (/) ) ) e. om ) )
16	15	simpld 475	. . . . . 6 |- ( ph -> _E We ( F supp (/) ) )
17	4	oiiso 8442	. . . . . 6 |- ( ( ( F supp (/) ) e. _V /\ _E We ( F supp (/) ) ) -> OrdIso ( _E , ( F supp (/) ) ) Isom _E , _E ( dom OrdIso ( _E , ( F supp (/) ) ) , ( F supp (/) ) ) )
18	11, 16, 17	syl2anc 693	. . . . 5 |- ( ph -> OrdIso ( _E , ( F supp (/) ) ) Isom _E , _E ( dom OrdIso ( _E , ( F supp (/) ) ) , ( F supp (/) ) ) )
19		isof1o 6573	. . . . 5 |- (OrdIso ( _E , ( F supp (/) ) ) Isom _E , _E ( dom OrdIso ( _E , ( F supp (/) ) ) , ( F supp (/) ) ) -> OrdIso ( _E , ( F supp (/) ) ) : dom OrdIso ( _E , ( F supp (/) ) ) -1-1-onto-> ( F supp (/) ) )
20		f1ofo 6144	. . . . 5 |- (OrdIso ( _E , ( F supp (/) ) ) : dom OrdIso ( _E , ( F supp (/) ) ) -1-1-onto-> ( F supp (/) ) -> OrdIso ( _E , ( F supp (/) ) ) : dom OrdIso ( _E , ( F supp (/) ) ) -onto-> ( F supp (/) ) )
21		foima 6120	. . . . 5 |- (OrdIso ( _E , ( F supp (/) ) ) : dom OrdIso ( _E , ( F supp (/) ) ) -onto-> ( F supp (/) ) -> (OrdIso ( _E , ( F supp (/) ) ) " dom OrdIso ( _E , ( F supp (/) ) ) ) = ( F supp (/) ) )
22	18, 19, 20, 21	4syl 19	. . . 4 |- ( ph -> (OrdIso ( _E , ( F supp (/) ) ) " dom OrdIso ( _E , ( F supp (/) ) ) ) = ( F supp (/) ) )
23	22, 10	eqsstrd 3639	. . 3 |- ( ph -> (OrdIso ( _E , ( F supp (/) ) ) " dom OrdIso ( _E , ( F supp (/) ) ) ) C_ C )
24	1, 2, 3, 4, 5, 6, 8, 14, 9, 23	cantnflt 8569	. 2 |- ( ph -> (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( F supp (/) ) ) ` k ) ) .o ( F ` (OrdIso ( _E , ( F supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` dom OrdIso ( _E , ( F supp (/) ) ) ) e. ( A ^o C ) )
25	7, 24	eqeltrd 2701	1 |- ( ph -> ( ( A CNF B ) ` F ) e. ( A ^o C ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   _E cep 5028   We wwe 5072   dom cdm 5114   "cima 5117   Oncon0 5723   suc csuc 5725   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   supp csupp 7295  seq_𝜔cseqom 7542   +o coa 7557   .o comu 7558   ^o coe 7559  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-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:  cantnff  8571  cantnflem1d  8585  cnfcom3lem  8600