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 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}
cantnffval.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnffval.b	⊢ (𝜑 → 𝐵 ∈ 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 ℎ 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnffval.s	. . . 4 ⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}
2		cantnffval.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ On)
3		cantnffval.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ On)
4	1, 2, 3	cantnffval 9701	. . 3 ⊢ (𝜑 → (𝐴 CNF 𝐵) = (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)))
5	4	dmeqd 5899	. 2 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = dom (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)))
6		fvex 6929	. . . . 5 ⊢ (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ) ∈ V
7	6	csbex 5271	. . . 4 ⊢ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ) ∈ V
8	7	rgenw 3084	. . 3 ⊢ ∀𝑓 ∈ 𝑆 ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ) ∈ V
9		dmmptg 6262	. . 3 ⊢ (∀𝑓 ∈ 𝑆 ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ) ∈ V → dom (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)) = 𝑆)
10	8, 9	ax-mp 5	. 2 ⊢ dom (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)) = 𝑆
11	5, 10	eqtrdi 2814	1 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454  ⦋csb 3851  ∅c0 4281   class class class wbr 5108   ↦ cmpt 5192   E cep 5557  dom cdm 5658  Oncon0 6388  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453   supp csupp 8214  seq_ωcseqom 8495   +_o coa 8511   ·_o comu 8512   ↑_o coe 8513   ↑_m cmap 8885   finSupp cfsupp 9391  OrdIsocoi 9540   CNF ccnf 9699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-seqom 8496  df-cnf 9700

This theorem is used by:  cantnfs  9704  cantnfval  9706  cantnff  9712  oemapso  9720  wemapwe  9735  oef1o  9736