Theorem cantnfvalf 9601

Description: Lemma for cantnf 9629. The function appearing in cantnfval 9604 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 8401	. 2 ⊢ 𝐹 Fn ω
3		nn0suc 7832	. . . 4 ⊢ (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅))
5		0ex 5264	. . . . . . . 8 ⊢ ∅ ∈ V
6	1	seqom0g 8402	. . . . . . . 8 ⊢ (∅ ∈ V → (𝐹‘∅) = ∅)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
8	4, 7	eqtrdi 2792	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = ∅)
9		0elon 6371	. . . . . 6 ⊢ ∅ ∈ On
10	8, 9	eqeltrdi 2846	. . . . 5 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) ∈ On)
11	1	seqomsuc 8403	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))(𝐹‘𝑦)))
12		df-ov 7360	. . . . . . . . 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 2792	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹‘𝑦)⟩))
14		df-ov 7360	. . . . . . . . . . . 12 ⊢ (𝐶 +o 𝐷) = ( +o ‘⟨𝐶, 𝐷⟩)
15		fnoa 8454	. . . . . . . . . . . . . 14 ⊢ +o Fn (On × On)
16		oacl 8481	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +o 𝑦) ∈ On)
17	16	rgen2 3194	. . . . . . . . . . . . . 14 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On
18		ffnov 7483	. . . . . . . . . . . . . 14 ⊢ ( +o :(On × On)⟶On ↔ ( +o Fn (On × On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On))
19	15, 17, 18	mpbir2an 709	. . . . . . . . . . . . 13 ⊢ +o :(On × On)⟶On
20	19, 9	f0cli 7048	. . . . . . . . . . . 12 ⊢ ( +o ‘⟨𝐶, 𝐷⟩) ∈ On
21	14, 20	eqeltri 2834	. . . . . . . . . . 11 ⊢ (𝐶 +o 𝐷) ∈ On
22	21	rgen2w 3069	. . . . . . . . . 10 ⊢ ∀𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐵 (𝐶 +o 𝐷) ∈ On
23		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)) = (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))
24	23	fmpo 8000	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐵 (𝐶 +o 𝐷) ∈ On ↔ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On)
25	22, 24	mpbi 229	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On
26	25, 9	f0cli 7048	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹‘𝑦)⟩) ∈ On
27	13, 26	eqeltrdi 2846	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28		fveq2 6842	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝐹‘𝑥) = (𝐹‘suc 𝑦))
29	28	eleq1d 2822	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝐹‘𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On))
30	27, 29	syl5ibrcom 246	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹‘𝑥) ∈ On))
31	30	rexlimiv 3145	. . . . 5 ⊢ (∃𝑦 ∈ ω 𝑥 = suc 𝑦 → (𝐹‘𝑥) ∈ On)
32	10, 31	jaoi 855	. . . 4 ⊢ ((𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦) → (𝐹‘𝑥) ∈ On)
33	3, 32	syl 17	. . 3 ⊢ (𝑥 ∈ ω → (𝐹‘𝑥) ∈ On)
34	33	rgen 3066	. 2 ⊢ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ On
35		ffnfv 7066	. 2 ⊢ (𝐹:ω⟶On ↔ (𝐹 Fn ω ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ On))
36	2, 34, 35	mpbir2an 709	1 ⊢ 𝐹:ω⟶On

Syntax hints:   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  Vcvv 3445  ∅c0 4282  ⟨cop 4592   × cxp 5631  Oncon0 6317  suc csuc 6319   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  ωcom 7802  seq_ωcseqom 8393   +_o coa 8409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-seqom 8394  df-oadd 8416

This theorem is referenced by:  cantnfval2  9605  cantnfle  9607  cantnflt  9608  cantnflem1d  9624  cantnflem1  9625  cnfcomlem  9635