Theorem cantnfvalf 8963

Description: Lemma for cantnf 8991. The function appearing in cantnfval 8966 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 7933	. 2 ⊢ 𝐹 Fn ω
3		nn0suc 7453	. . . 4 ⊢ (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4		fveq2 6530	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅))
5		0ex 5096	. . . . . . . 8 ⊢ ∅ ∈ V
6	1	seqom0g 7934	. . . . . . . 8 ⊢ (∅ ∈ V → (𝐹‘∅) = ∅)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
8	4, 7	syl6eq 2845	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = ∅)
9		0elon 6111	. . . . . 6 ⊢ ∅ ∈ On
10	8, 9	syl6eqel 2889	. . . . 5 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) ∈ On)
11	1	seqomsuc 7935	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))(𝐹‘𝑦)))
12		df-ov 7010	. . . . . . . . 9 ⊢ (𝑦(𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))(𝐹‘𝑦)) = ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹‘𝑦)⟩)
13	11, 12	syl6eq 2845	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹‘𝑦)⟩))
14		df-ov 7010	. . . . . . . . . . . 12 ⊢ (𝐶 +o 𝐷) = ( +o ‘⟨𝐶, 𝐷⟩)
15		fnoa 7975	. . . . . . . . . . . . . 14 ⊢ +o Fn (On × On)
16		oacl 8002	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +o 𝑦) ∈ On)
17	16	rgen2a 3191	. . . . . . . . . . . . . 14 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On
18		ffnov 7125	. . . . . . . . . . . . . 14 ⊢ ( +o :(On × On)⟶On ↔ ( +o Fn (On × On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On))
19	15, 17, 18	mpbir2an 707	. . . . . . . . . . . . 13 ⊢ +o :(On × On)⟶On
20	19, 9	f0cli 6718	. . . . . . . . . . . 12 ⊢ ( +o ‘⟨𝐶, 𝐷⟩) ∈ On
21	14, 20	eqeltri 2877	. . . . . . . . . . 11 ⊢ (𝐶 +o 𝐷) ∈ On
22	21	rgen2w 3116	. . . . . . . . . 10 ⊢ ∀𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐵 (𝐶 +o 𝐷) ∈ On
23		eqid 2793	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)) = (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))
24	23	fmpo 7613	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐵 (𝐶 +o 𝐷) ∈ On ↔ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On)
25	22, 24	mpbi 231	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On
26	25, 9	f0cli 6718	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹‘𝑦)⟩) ∈ On
27	13, 26	syl6eqel 2889	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28		fveq2 6530	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝐹‘𝑥) = (𝐹‘suc 𝑦))
29	28	eleq1d 2865	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝐹‘𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On))
30	27, 29	syl5ibrcom 248	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹‘𝑥) ∈ On))
31	30	rexlimiv 3240	. . . . 5 ⊢ (∃𝑦 ∈ ω 𝑥 = suc 𝑦 → (𝐹‘𝑥) ∈ On)
32	10, 31	jaoi 852	. . . 4 ⊢ ((𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦) → (𝐹‘𝑥) ∈ On)
33	3, 32	syl 17	. . 3 ⊢ (𝑥 ∈ ω → (𝐹‘𝑥) ∈ On)
34	33	rgen 3113	. 2 ⊢ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ On
35		ffnfv 6736	. 2 ⊢ (𝐹:ω⟶On ↔ (𝐹 Fn ω ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ On))
36	2, 34, 35	mpbir2an 707	1 ⊢ 𝐹:ω⟶On

Syntax hints:   ∨ wo 842   = wceq 1520   ∈ wcel 2079  ∀wral 3103  ∃wrex 3104  Vcvv 3432  ∅c0 4206  ⟨cop 4472   × cxp 5433  Oncon0 6058  suc csuc 6060   Fn wfn 6212  ⟶wf 6213  ‘cfv 6217  (class class class)co 7007   ∈ cmpo 7009  ωcom 7427  seq_𝜔cseqom 7925   +_o coa 7941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-seqom 7926  df-oadd 7948

This theorem is referenced by:  cantnfval2  8967  cantnfle  8969  cantnflt  8970  cantnflem1d  8986  cantnflem1  8987  cnfcomlem  8997