Theorem cantnfdm 9617

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 9616	. . 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 5869	. 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 6871	. . . . 5 ⊢ (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ) ∈ V
7	6	csbex 5266	. . . 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 3048	. . 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 6215	. . 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 2780	1 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447  ⦋csb 3862  ∅c0 4296   class class class wbr 5107   ↦ cmpt 5188   E cep 5537  dom cdm 5638  Oncon0 6332  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389   supp csupp 8139  seq_ωcseqom 8415   +_o coa 8431   ·_o comu 8432   ↑_o coe 8433   ↑_m cmap 8799   finSupp cfsupp 9312  OrdIsocoi 9462   CNF ccnf 9614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-seqom 8416  df-cnf 9615

This theorem is referenced by:  cantnfs  9619  cantnfval  9621  cantnff  9627  oemapso  9635  wemapwe  9650  oef1o  9651