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Theorem cantnfdm 8838
 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 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅}
cantnffval.a (𝜑𝐴 ∈ On)
cantnffval.b (𝜑𝐵 ∈ On)
Assertion
Ref Expression
cantnfdm (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆)
Distinct variable groups:   𝐴,𝑔   𝐵,𝑔
Allowed substitution hints:   𝜑(𝑔)   𝑆(𝑔)

Proof of Theorem cantnfdm
Dummy variables 𝑓 𝑘 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnffval.s . . . 4 𝑆 = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅}
2 cantnffval.a . . . 4 (𝜑𝐴 ∈ On)
3 cantnffval.b . . . 4 (𝜑𝐵 ∈ On)
41, 2, 3cantnffval 8837 . . 3 (𝜑 → (𝐴 CNF 𝐵) = (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom )))
54dmeqd 5558 . 2 (𝜑 → dom (𝐴 CNF 𝐵) = dom (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom )))
6 fvex 6446 . . . . 5 (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ) ∈ V
76csbex 5018 . . . 4 OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ) ∈ V
87rgenw 3133 . . 3 𝑓𝑆 OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ) ∈ V
9 dmmptg 5873 . . 3 (∀𝑓𝑆 OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ) ∈ V → dom (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom )) = 𝑆)
108, 9ax-mp 5 . 2 dom (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom )) = 𝑆
115, 10syl6eq 2877 1 (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  ∀wral 3117  {crab 3121  Vcvv 3414  ⦋csb 3757  ∅c0 4144   class class class wbr 4873   ↦ cmpt 4952   E cep 5254  dom cdm 5342  Oncon0 5963  ‘cfv 6123  (class class class)co 6905   ↦ cmpt2 6907   supp csupp 7559  seq𝜔cseqom 7808   +o coa 7823   ·o comu 7824   ↑o coe 7825   ↑𝑚 cmap 8122   finSupp cfsupp 8544  OrdIsocoi 8683   CNF ccnf 8835 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-seqom 7809  df-cnf 8836 This theorem is referenced by:  cantnfs  8840  cantnfval  8842  cantnff  8848  oemapso  8856  wemapwe  8871  oef1o  8872
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