Theorem cantnfvalf 8445

Description: Lemma for cantnf 8473. The function appearing in cantnfval 8448 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 7437	. 2 ⊢ 𝐹 Fn ω
3		nn0suc 6982	. . . 4 ⊢ (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4		fveq2 6103	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅))
5		0ex 4718	. . . . . . . 8 ⊢ ∅ ∈ V
6	1	seqom0g 7438	. . . . . . . 8 ⊢ (∅ ∈ V → (𝐹‘∅) = ∅)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
8	4, 7	syl6eq 2660	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = ∅)
9		0elon 5695	. . . . . 6 ⊢ ∅ ∈ On
10	8, 9	syl6eqel 2696	. . . . 5 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) ∈ On)
11	1	seqomsuc 7439	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +𝑜 𝐷))(𝐹‘𝑦)))
12		df-ov 6552	. . . . . . . . 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 2660	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +𝑜 𝐷))‘⟨𝑦, (𝐹‘𝑦)⟩))
14		df-ov 6552	. . . . . . . . . . . 12 ⊢ (𝐶 +𝑜 𝐷) = ( +𝑜 ‘⟨𝐶, 𝐷⟩)
15		fnoa 7475	. . . . . . . . . . . . . 14 ⊢ +𝑜 Fn (On × On)
16		oacl 7502	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +𝑜 𝑦) ∈ On)
17	16	rgen2a 2960	. . . . . . . . . . . . . 14 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +𝑜 𝑦) ∈ On
18		ffnov 6662	. . . . . . . . . . . . . 14 ⊢ ( +𝑜 :(On × On)⟶On ↔ ( +𝑜 Fn (On × On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +𝑜 𝑦) ∈ On))
19	15, 17, 18	mpbir2an 957	. . . . . . . . . . . . 13 ⊢ +𝑜 :(On × On)⟶On
20	19, 9	f0cli 6278	. . . . . . . . . . . 12 ⊢ ( +𝑜 ‘⟨𝐶, 𝐷⟩) ∈ On
21	14, 20	eqeltri 2684	. . . . . . . . . . 11 ⊢ (𝐶 +𝑜 𝐷) ∈ On
22	21	rgen2w 2909	. . . . . . . . . 10 ⊢ ∀𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐵 (𝐶 +𝑜 𝐷) ∈ On
23		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +𝑜 𝐷)) = (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +𝑜 𝐷))
24	23	fmpt2 7126	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐵 (𝐶 +𝑜 𝐷) ∈ On ↔ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +𝑜 𝐷)):(𝐴 × 𝐵)⟶On)
25	22, 24	mpbi 219	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +𝑜 𝐷)):(𝐴 × 𝐵)⟶On
26	25, 9	f0cli 6278	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +𝑜 𝐷))‘⟨𝑦, (𝐹‘𝑦)⟩) ∈ On
27	13, 26	syl6eqel 2696	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝐹‘𝑥) = (𝐹‘suc 𝑦))
29	28	eleq1d 2672	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝐹‘𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On))
30	27, 29	syl5ibrcom 236	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹‘𝑥) ∈ On))
31	30	rexlimiv 3009	. . . . 5 ⊢ (∃𝑦 ∈ ω 𝑥 = suc 𝑦 → (𝐹‘𝑥) ∈ On)
32	10, 31	jaoi 393	. . . 4 ⊢ ((𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦) → (𝐹‘𝑥) ∈ On)
33	3, 32	syl 17	. . 3 ⊢ (𝑥 ∈ ω → (𝐹‘𝑥) ∈ On)
34	33	rgen 2906	. 2 ⊢ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ On
35		ffnfv 6295	. 2 ⊢ (𝐹:ω⟶On ↔ (𝐹 Fn ω ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ On))
36	2, 34, 35	mpbir2an 957	1 ⊢ 𝐹:ω⟶On

Syntax hints:   ∨ wo 382   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ∅c0 3874  ⟨cop 4131   × cxp 5036  Oncon0 5640  suc csuc 5642   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  seq_𝜔cseqom 7429   +_𝑜 coa 7444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-oadd 7451

This theorem is referenced by:  cantnfval2  8449  cantnfle  8451  cantnflt  8452  cantnflem1d  8468  cantnflem1  8469  cnfcomlem  8479