Theorem cantnfdm 8561

Description: The domain of the Cantor normal form function (in later lemmas we will use dom ( A CNF B ) to abbreviate "the set of finitely supported functions from B to A"). (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)

Hypotheses

Ref	Expression
cantnffval.s	|- S = { g e. ( A ^m B ) | g finSupp (/) }
cantnffval.a	|- ( ph -> A e. On )
cantnffval.b	|- ( ph -> B e. On )

Assertion

Ref	Expression
cantnfdm	|- ( ph -> dom ( A CNF B ) = S )

Distinct variable groups:   A, g   B, g

Allowed substitution hints:   ph( g)   S( g)

Proof of Theorem cantnfdm

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

Step	Hyp	Ref	Expression
1		cantnffval.s	. . . 4 |- S = { g e. ( A ^m B ) | g finSupp (/) }
2		cantnffval.a	. . . 4 |- ( ph -> A e. On )
3		cantnffval.b	. . . 4 |- ( ph -> B e. On )
4	1, 2, 3	cantnffval 8560	. . 3 |- ( ph -> ( A CNF B ) = ( f e. S |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) )
5	4	dmeqd 5326	. 2 |- ( ph -> dom ( A CNF B ) = dom ( f e. S |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) )
6		fvex 6201	. . . . 5 |- (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) e. _V
7	6	csbex 4793	. . . 4 |- [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) e. _V
8	7	rgenw 2924	. . 3 |- A. f e. S [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) e. _V
9		dmmptg 5632	. . 3 |- ( A. f e. S [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) e. _V -> dom ( f e. S |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) = S )
10	8, 9	ax-mp 5	. 2 |- dom ( f e. S |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) = S
11	5, 10	syl6eq 2672	1 |- ( ph -> dom ( A CNF B ) = S )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   [_csb 3533   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   _E cep 5028   dom cdm 5114   Oncon0 5723   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   supp csupp 7295  seq_𝜔cseqom 7542   +o coa 7557   .o comu 7558   ^o coe 7559   ^m cmap 7857   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-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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-cnf 8559

This theorem is referenced by:  cantnfs  8563  cantnfval  8565  cantnff  8571  oemapso  8579  wemapwe  8594  oef1o  8595