Theorem cantnfvalf 9578

Description: Lemma for cantnf 9606. The function appearing in cantnfval 9581 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 8388	. 2 ⊢ 𝐹 Fn ω
3		nn0suc 7838	. . . 4 ⊢ (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4		fveq2 6835	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅))
5		0ex 5253	. . . . . . . 8 ⊢ ∅ ∈ V
6	1	seqom0g 8389	. . . . . . . 8 ⊢ (∅ ∈ V → (𝐹‘∅) = ∅)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
8	4, 7	eqtrdi 2788	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = ∅)
9		0elon 6373	. . . . . 6 ⊢ ∅ ∈ On
10	8, 9	eqeltrdi 2845	. . . . 5 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) ∈ On)
11	1	seqomsuc 8390	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))(𝐹‘𝑦)))
12		df-ov 7363	. . . . . . . . 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 2788	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹‘𝑦)⟩))
14		df-ov 7363	. . . . . . . . . . . 12 ⊢ (𝐶 +o 𝐷) = ( +o ‘⟨𝐶, 𝐷⟩)
15		fnoa 8437	. . . . . . . . . . . . . 14 ⊢ +o Fn (On × On)
16		oacl 8464	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +o 𝑦) ∈ On)
17	16	rgen2 3177	. . . . . . . . . . . . . 14 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On
18		ffnov 7486	. . . . . . . . . . . . . 14 ⊢ ( +o :(On × On)⟶On ↔ ( +o Fn (On × On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On))
19	15, 17, 18	mpbir2an 712	. . . . . . . . . . . . 13 ⊢ +o :(On × On)⟶On
20	19, 9	f0cli 7045	. . . . . . . . . . . 12 ⊢ ( +o ‘⟨𝐶, 𝐷⟩) ∈ On
21	14, 20	eqeltri 2833	. . . . . . . . . . 11 ⊢ (𝐶 +o 𝐷) ∈ On
22	21	rgen2w 3057	. . . . . . . . . 10 ⊢ ∀𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐵 (𝐶 +o 𝐷) ∈ On
23		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)) = (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))
24	23	fmpo 8014	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐵 (𝐶 +o 𝐷) ∈ On ↔ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On)
25	22, 24	mpbi 230	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On
26	25, 9	f0cli 7045	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹‘𝑦)⟩) ∈ On
27	13, 26	eqeltrdi 2845	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28		fveq2 6835	. . . . . . . 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 247	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹‘𝑥) ∈ On))
31	30	rexlimiv 3131	. . . . 5 ⊢ (∃𝑦 ∈ ω 𝑥 = suc 𝑦 → (𝐹‘𝑥) ∈ On)
32	10, 31	jaoi 858	. . . 4 ⊢ ((𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦) → (𝐹‘𝑥) ∈ On)
33	3, 32	syl 17	. . 3 ⊢ (𝑥 ∈ ω → (𝐹‘𝑥) ∈ On)
34	33	rgen 3054	. 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 712	1 ⊢ 𝐹:ω⟶On

Syntax hints:   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061  Vcvv 3441  ∅c0 4286  ⟨cop 4587   × cxp 5623  Oncon0 6318  suc csuc 6320   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360   ∈ cmpo 7362  ωcom 7810  seq_ωcseqom 8380   +_o coa 8396

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-seqom 8381  df-oadd 8403

This theorem is referenced by:  cantnfval2  9582  cantnfle  9584  cantnflt  9585  cantnflem1d  9601  cantnflem1  9602  cnfcomlem  9612