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Theorem cantnfdm 8546
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 8545 . . 3 (𝜑 → (𝐴 CNF 𝐵) = (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )))
54dmeqd 5315 . 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 6188 . . . . 5 (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom ) ∈ V
76csbex 4784 . . . 4 OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom ) ∈ V
87rgenw 2921 . . 3 𝑓𝑆 OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom ) ∈ V
9 dmmptg 5620 . . 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 2670 1 (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  wral 2909  {crab 2913  Vcvv 3195  csb 3526  c0 3907   class class class wbr 4644  cmpt 4720   E cep 5018  dom cdm 5104  Oncon0 5711  cfv 5876  (class class class)co 6635  cmpt2 6637   supp csupp 7280  seq𝜔cseqom 7527   +𝑜 coa 7542   ·𝑜 comu 7543  𝑜 coe 7544  𝑚 cmap 7842   finSupp cfsupp 8260  OrdIsocoi 8399   CNF ccnf 8543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seqom 7528  df-cnf 8544
This theorem is referenced by:  cantnfs  8548  cantnfval  8550  cantnff  8556  oemapso  8564  wemapwe  8579  oef1o  8580
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