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Theorem cantnfdm 9702
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 9701 . . 3 (𝜑 → (𝐴 CNF 𝐵) = (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom )))
54dmeqd 5919 . 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 6920 . . . . 5 (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ) ∈ V
76csbex 5317 . . . 4 OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ) ∈ V
87rgenw 3063 . . 3 𝑓𝑆 OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ) ∈ V
9 dmmptg 6264 . . 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, 10eqtrdi 2791 1 (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  csb 3908  c0 4339   class class class wbr 5148  cmpt 5231   E cep 5588  dom cdm 5689  Oncon0 6386  cfv 6563  (class class class)co 7431  cmpo 7433   supp csupp 8184  seqωcseqom 8486   +o coa 8502   ·o comu 8503  o coe 8504  m cmap 8865   finSupp cfsupp 9399  OrdIsocoi 9547   CNF ccnf 9699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-cnf 9700
This theorem is referenced by:  cantnfs  9704  cantnfval  9706  cantnff  9712  oemapso  9720  wemapwe  9735  oef1o  9736
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