Theorem cantnfvalf 9634

Description: Lemma for cantnf 9662. The function appearing in cantnfval 9637 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)

Hypothesis

Ref	Expression
cantnfvalf.f	⊢ 𝐹 = seqω((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)), ∅)

Assertion

Ref	Expression
cantnfvalf	⊢ 𝐹:ω⟶On

Distinct variable groups:   𝑧,𝑘,𝐴   𝐵,𝑘,𝑧

Allowed substitution hints:   𝐶(𝑧,𝑘)   𝐷(𝑧,𝑘)   𝐹(𝑧,𝑘)

Proof of Theorem cantnfvalf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfvalf.f	. . 3 ⊢ 𝐹 = seqω((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)), ∅)
2	1	fnseqom 8442	. 2 ⊢ 𝐹 Fn ω
3		nn0suc 7891	. . . 4 ⊢ (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4		fveq2 6882	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅))
5		0ex 5272	. . . . . . . 8 ⊢ ∅ ∈ V
6	1	seqom0g 8443	. . . . . . . 8 ⊢ (∅ ∈ V → (𝐹‘∅) = ∅)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
8	4, 7	eqtrdi 2820	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = ∅)
9		0elon 6417	. . . . . 6 ⊢ ∅ ∈ On
10	8, 9	eqeltrdi 2877	. . . . 5 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) ∈ On)
11	1	seqomsuc 8444	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))(𝐹‘𝑦)))
12		df-ov 7414	. . . . . . . . 9 ⊢ (𝑦(𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))(𝐹‘𝑦)) = ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹‘𝑦)⟩)
13	11, 12	eqtrdi 2820	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹‘𝑦)⟩))
14		df-ov 7414	. . . . . . . . . . . 12 ⊢ (𝐶 +o 𝐷) = ( +o ‘⟨𝐶, 𝐷⟩)
15		fnoa 8493	. . . . . . . . . . . . . 14 ⊢ +o Fn (On × On)
16		oacl 8520	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +o 𝑦) ∈ On)
17	16	rgen2 3211	. . . . . . . . . . . . . 14 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On
18		ffnov 7537	. . . . . . . . . . . . . 14 ⊢ ( +o :(On × On)⟶On ↔ ( +o Fn (On × On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On))
19	15, 17, 18	mpbir2an 723	. . . . . . . . . . . . 13 ⊢ +o :(On × On)⟶On
20	19, 9	f0cli 7094	. . . . . . . . . . . 12 ⊢ ( +o ‘⟨𝐶, 𝐷⟩) ∈ On
21	14, 20	eqeltri 2865	. . . . . . . . . . 11 ⊢ (𝐶 +o 𝐷) ∈ On
22	21	rgen2w 3090	. . . . . . . . . 10 ⊢ ∀𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐵 (𝐶 +o 𝐷) ∈ On
23		eqid 2769	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)) = (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))
24	23	fmpo 8065	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐵 (𝐶 +o 𝐷) ∈ On ↔ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On)
25	22, 24	mpbi 233	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On
26	25, 9	f0cli 7094	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹‘𝑦)⟩) ∈ On
27	13, 26	eqeltrdi 2877	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝐹‘𝑥) = (𝐹‘suc 𝑦))
29	28	eleq1d 2854	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝐹‘𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On))
30	27, 29	syl5ibrcom 250	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹‘𝑥) ∈ On))
31	30	rexlimiv 3165	. . . . 5 ⊢ (∃𝑦 ∈ ω 𝑥 = suc 𝑦 → (𝐹‘𝑥) ∈ On)
32	10, 31	jaoi 870	. . . 4 ⊢ ((𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦) → (𝐹‘𝑥) ∈ On)
33	3, 32	syl 18	. . 3 ⊢ (𝑥 ∈ ω → (𝐹‘𝑥) ∈ On)
34	33	rgen 3087	. 2 ⊢ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ On
35		ffnfv 7115	. 2 ⊢ (𝐹:ω⟶On ↔ (𝐹 Fn ω ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ On))
36	2, 34, 35	mpbir2an 723	1 ⊢ 𝐹:ω⟶On

Syntax hints:   ∨ wo 860   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  Vcvv 3463  ∅c0 4294  ⟨cop 4600   × cxp 5660  Oncon0 6361  suc csuc 6363   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  ωcom 7862  seq_ωcseqom 8434   +_o coa 8450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-seqom 8435  df-oadd 8457

This theorem is referenced by:  cantnfval2  9638  cantnfle  9640  cantnflt  9641  cantnflem1d  9657  cantnflem1  9658  cnfcomlem  9668