Theorem cantnflem1d 8835

Description: Lemma for cantnf 8840. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
oemapval.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
oemapval.f	⊢ (𝜑 → 𝐹 ∈ 𝑆)
oemapval.g	⊢ (𝜑 → 𝐺 ∈ 𝑆)
oemapvali.r	⊢ (𝜑 → 𝐹𝑇𝐺)
oemapvali.x	⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}
cantnflem1.o	⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))
cantnflem1.h	⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝑂‘𝑘)) ·𝑜 (𝐺‘(𝑂‘𝑘))) +𝑜 𝑧)), ∅)

Assertion

Ref	Expression
cantnflem1d	⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ (𝐻‘suc (◡𝑂‘𝑋)))

Distinct variable groups:   𝑘,𝑐,𝑤,𝑥,𝑦,𝑧,𝐵   𝐴,𝑐,𝑘,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐,𝑘   𝑘,𝐹,𝑤,𝑥,𝑦,𝑧   𝑆,𝑐,𝑘,𝑥,𝑦,𝑧   𝐺,𝑐,𝑘,𝑤,𝑥,𝑦,𝑧   𝑥,𝐻,𝑦   𝑘,𝑂,𝑤,𝑥,𝑦,𝑧   𝜑,𝑘,𝑥,𝑦,𝑧   𝑘,𝑋,𝑤,𝑥,𝑦,𝑧   𝐹,𝑐   𝜑,𝑐

Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝐻(𝑧,𝑤,𝑘,𝑐)   𝑂(𝑐)   𝑋(𝑐)

Proof of Theorem cantnflem1d

Step	Hyp	Ref	Expression
1		cantnfs.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ On)
2		cantnfs.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ On)
3		cantnfs.s	. . . . . . . . 9 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
4		oemapval.t	. . . . . . . . 9 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
5		oemapval.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
6		oemapval.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝑆)
7		oemapvali.r	. . . . . . . . 9 ⊢ (𝜑 → 𝐹𝑇𝐺)
8		oemapvali.x	. . . . . . . . 9 ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}
9	3, 1, 2, 4, 5, 6, 7, 8	oemapvali 8831	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
10	9	simp1d 1165	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		onelon 5968	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
12	2, 10, 11	syl2anc 575	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On)
13		oecl 7857	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑𝑜 𝑋) ∈ On)
14	1, 12, 13	syl2anc 575	. . . . 5 ⊢ (𝜑 → (𝐴 ↑𝑜 𝑋) ∈ On)
15	3, 1, 2	cantnfs 8813	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
16	6, 15	mpbid 223	. . . . . . . 8 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
17	16	simpld 484	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
18	17, 10	ffvelrnd 6585	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
19		onelon 5968	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑋) ∈ 𝐴) → (𝐺‘𝑋) ∈ On)
20	1, 18, 19	syl2anc 575	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ On)
21		omcl 7856	. . . . 5 ⊢ (((𝐴 ↑𝑜 𝑋) ∈ On ∧ (𝐺‘𝑋) ∈ On) → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) ∈ On)
22	14, 20, 21	syl2anc 575	. . . 4 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) ∈ On)
23		suppssdm 7545	. . . . . . . . . . . 12 ⊢ (𝐺 supp ∅) ⊆ dom 𝐺
24	23, 17	fssdm 6275	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
25	2, 24	ssexd 5007	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 supp ∅) ∈ V)
26		cantnflem1.o	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))
27	3, 1, 2, 26, 6	cantnfcl 8814	. . . . . . . . . . 11 ⊢ (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
28	27	simpld 484	. . . . . . . . . 10 ⊢ (𝜑 → E We (𝐺 supp ∅))
29	26	oiiso 8684	. . . . . . . . . 10 ⊢ (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
30	25, 28, 29	syl2anc 575	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
31		isof1o 6800	. . . . . . . . 9 ⊢ (𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) → 𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅))
32	30, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅))
33		f1ocnv 6368	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) → ◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
34		f1of 6356	. . . . . . . 8 ⊢ (◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂 → ◡𝑂:(𝐺 supp ∅)⟶dom 𝑂)
35	32, 33, 34	3syl 18	. . . . . . 7 ⊢ (𝜑 → ◡𝑂:(𝐺 supp ∅)⟶dom 𝑂)
36	3, 1, 2, 4, 5, 6, 7, 8	cantnflem1a 8832	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))
37	35, 36	ffvelrnd 6585	. . . . . 6 ⊢ (𝜑 → (◡𝑂‘𝑋) ∈ dom 𝑂)
38	27	simprd 485	. . . . . 6 ⊢ (𝜑 → dom 𝑂 ∈ ω)
39		elnn 7308	. . . . . 6 ⊢ (((◡𝑂‘𝑋) ∈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → (◡𝑂‘𝑋) ∈ ω)
40	37, 38, 39	syl2anc 575	. . . . 5 ⊢ (𝜑 → (◡𝑂‘𝑋) ∈ ω)
41		cantnflem1.h	. . . . . . 7 ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝑂‘𝑘)) ·𝑜 (𝐺‘(𝑂‘𝑘))) +𝑜 𝑧)), ∅)
42	41	cantnfvalf 8812	. . . . . 6 ⊢ 𝐻:ω⟶On
43	42	ffvelrni 6583	. . . . 5 ⊢ ((◡𝑂‘𝑋) ∈ ω → (𝐻‘(◡𝑂‘𝑋)) ∈ On)
44	40, 43	syl 17	. . . 4 ⊢ (𝜑 → (𝐻‘(◡𝑂‘𝑋)) ∈ On)
45		oaword1 7872	. . . 4 ⊢ ((((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) ∈ On ∧ (𝐻‘(◡𝑂‘𝑋)) ∈ On) → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) ⊆ (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) +𝑜 (𝐻‘(◡𝑂‘𝑋))))
46	22, 44, 45	syl2anc 575	. . 3 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) ⊆ (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) +𝑜 (𝐻‘(◡𝑂‘𝑋))))
47	3, 1, 2, 26, 6, 41	cantnfsuc 8817	. . . . 5 ⊢ ((𝜑 ∧ (◡𝑂‘𝑋) ∈ ω) → (𝐻‘suc (◡𝑂‘𝑋)) = (((𝐴 ↑𝑜 (𝑂‘(◡𝑂‘𝑋))) ·𝑜 (𝐺‘(𝑂‘(◡𝑂‘𝑋)))) +𝑜 (𝐻‘(◡𝑂‘𝑋))))
48	40, 47	mpdan 670	. . . 4 ⊢ (𝜑 → (𝐻‘suc (◡𝑂‘𝑋)) = (((𝐴 ↑𝑜 (𝑂‘(◡𝑂‘𝑋))) ·𝑜 (𝐺‘(𝑂‘(◡𝑂‘𝑋)))) +𝑜 (𝐻‘(◡𝑂‘𝑋))))
49		f1ocnvfv2 6760	. . . . . . . 8 ⊢ ((𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(◡𝑂‘𝑋)) = 𝑋)
50	32, 36, 49	syl2anc 575	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(◡𝑂‘𝑋)) = 𝑋)
51	50	oveq2d 6893	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑𝑜 (𝑂‘(◡𝑂‘𝑋))) = (𝐴 ↑𝑜 𝑋))
52	50	fveq2d 6415	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑂‘(◡𝑂‘𝑋))) = (𝐺‘𝑋))
53	51, 52	oveq12d 6895	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑𝑜 (𝑂‘(◡𝑂‘𝑋))) ·𝑜 (𝐺‘(𝑂‘(◡𝑂‘𝑋)))) = ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)))
54	53	oveq1d 6892	. . . 4 ⊢ (𝜑 → (((𝐴 ↑𝑜 (𝑂‘(◡𝑂‘𝑋))) ·𝑜 (𝐺‘(𝑂‘(◡𝑂‘𝑋)))) +𝑜 (𝐻‘(◡𝑂‘𝑋))) = (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) +𝑜 (𝐻‘(◡𝑂‘𝑋))))
55	48, 54	eqtrd 2847	. . 3 ⊢ (𝜑 → (𝐻‘suc (◡𝑂‘𝑋)) = (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) +𝑜 (𝐻‘(◡𝑂‘𝑋))))
56	46, 55	sseqtr4d 3846	. 2 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) ⊆ (𝐻‘suc (◡𝑂‘𝑋)))
57		onss 7223	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
58	2, 57	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ On)
59	58	sselda 3805	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
60	12	adantr 468	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ On)
61		onsseleq 5984	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥 ⊆ 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
62	59, 60, 61	syl2anc 575	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
63		orcom 888	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋) ↔ (𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋))
64	62, 63	syl6bb 278	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝑋 ↔ (𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋)))
65	64	ifbid 4308	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅) = if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅))
66	65	mpteq2dva 4945	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅)) = (𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅)))
67	66	fveq2d 6415	. . . 4 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) = ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅))))
68	3, 1, 2	cantnfs 8813	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
69	5, 68	mpbid 223	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))
70	69	simpld 484	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
71	70	ffvelrnda 6584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ 𝐴)
72	18	ne0d 4130	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≠ ∅)
73		on0eln0 5999	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
74	1, 73	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
75	72, 74	mpbird 248	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ 𝐴)
76	75	adantr 468	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ 𝐴)
77	71, 76	ifcld 4331	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) ∈ 𝐴)
78	77	fmpttd 6610	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)):𝐵⟶𝐴)
79		0ex 4991	. . . . . . . . 9 ⊢ ∅ ∈ V
80	79	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ V)
81	69	simprd 485	. . . . . . . 8 ⊢ (𝜑 → 𝐹 finSupp ∅)
82	70, 2, 80, 81	fsuppmptif 8547	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) finSupp ∅)
83	3, 1, 2	cantnfs 8813	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) ∈ 𝑆 ↔ ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)):𝐵⟶𝐴 ∧ (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) finSupp ∅)))
84	78, 82, 83	mpbir2and 695	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) ∈ 𝑆)
85	70, 10	ffvelrnd 6585	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐴)
86		eldifn 3939	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∖ 𝑋) → ¬ 𝑦 ∈ 𝑋)
87	86	adantl 469	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ 𝑋)) → ¬ 𝑦 ∈ 𝑋)
88	87	iffalsed 4297	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ 𝑋)) → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) = ∅)
89	88, 2	suppss2 7567	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) supp ∅) ⊆ 𝑋)
90		fveq2 6411	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
91	90	adantl 469	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
92	91	ifeq1da 4316	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → if(𝑥 = 𝑋, (𝐹‘𝑥), ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹‘𝑋), ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥)))
93		eleq1w 2875	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋))
94		fveq2 6411	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
95	93, 94	ifbieq1d 4309	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) = if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
96		eqid 2813	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) = (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))
97		fvex 6424	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑥) ∈ V
98	97, 79	ifex 4334	. . . . . . . . . . 11 ⊢ if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅) ∈ V
99	95, 96, 98	fvmpt 6506	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥) = if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
100	99	ifeq2d 4305	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → if(𝑥 = 𝑋, (𝐹‘𝑥), ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹‘𝑥), if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅)))
101	92, 100	eqtr3d 2849	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → if(𝑥 = 𝑋, (𝐹‘𝑋), ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹‘𝑥), if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅)))
102		ifor 4338	. . . . . . . 8 ⊢ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅) = if(𝑥 = 𝑋, (𝐹‘𝑥), if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
103	101, 102	syl6reqr 2866	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅) = if(𝑥 = 𝑋, (𝐹‘𝑋), ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥)))
104	103	mpteq2ia 4941	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅)) = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 𝑋, (𝐹‘𝑋), ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥)))
105	3, 1, 2, 84, 10, 85, 89, 104	cantnfp1 8828	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅)) ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅))) = (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))))))
106	105	simprd 485	. . . 4 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅))) = (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)))))
107	67, 106	eqtrd 2847	. . 3 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) = (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)))))
108		onelon 5968	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐹‘𝑋) ∈ 𝐴) → (𝐹‘𝑋) ∈ On)
109	1, 85, 108	syl2anc 575	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ On)
110		omsuc 7846	. . . . . 6 ⊢ (((𝐴 ↑𝑜 𝑋) ∈ On ∧ (𝐹‘𝑋) ∈ On) → ((𝐴 ↑𝑜 𝑋) ·𝑜 suc (𝐹‘𝑋)) = (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 (𝐴 ↑𝑜 𝑋)))
111	14, 109, 110	syl2anc 575	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 suc (𝐹‘𝑋)) = (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 (𝐴 ↑𝑜 𝑋)))
112		eloni 5953	. . . . . . . 8 ⊢ ((𝐺‘𝑋) ∈ On → Ord (𝐺‘𝑋))
113	20, 112	syl 17	. . . . . . 7 ⊢ (𝜑 → Ord (𝐺‘𝑋))
114	9	simp2d 1166	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐺‘𝑋))
115		ordsucss 7251	. . . . . . 7 ⊢ (Ord (𝐺‘𝑋) → ((𝐹‘𝑋) ∈ (𝐺‘𝑋) → suc (𝐹‘𝑋) ⊆ (𝐺‘𝑋)))
116	113, 114, 115	sylc 65	. . . . . 6 ⊢ (𝜑 → suc (𝐹‘𝑋) ⊆ (𝐺‘𝑋))
117		suceloni 7246	. . . . . . . 8 ⊢ ((𝐹‘𝑋) ∈ On → suc (𝐹‘𝑋) ∈ On)
118	109, 117	syl 17	. . . . . . 7 ⊢ (𝜑 → suc (𝐹‘𝑋) ∈ On)
119		omwordi 7891	. . . . . . 7 ⊢ ((suc (𝐹‘𝑋) ∈ On ∧ (𝐺‘𝑋) ∈ On ∧ (𝐴 ↑𝑜 𝑋) ∈ On) → (suc (𝐹‘𝑋) ⊆ (𝐺‘𝑋) → ((𝐴 ↑𝑜 𝑋) ·𝑜 suc (𝐹‘𝑋)) ⊆ ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋))))
120	118, 20, 14, 119	syl3anc 1483	. . . . . 6 ⊢ (𝜑 → (suc (𝐹‘𝑋) ⊆ (𝐺‘𝑋) → ((𝐴 ↑𝑜 𝑋) ·𝑜 suc (𝐹‘𝑋)) ⊆ ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋))))
121	116, 120	mpd 15	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 suc (𝐹‘𝑋)) ⊆ ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)))
122	111, 121	eqsstr3d 3844	. . . 4 ⊢ (𝜑 → (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 (𝐴 ↑𝑜 𝑋)) ⊆ ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)))
123	3, 1, 2, 84, 75, 12, 89	cantnflt2 8820	. . . . 5 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ (𝐴 ↑𝑜 𝑋))
124		onelon 5968	. . . . . . 7 ⊢ (((𝐴 ↑𝑜 𝑋) ∈ On ∧ ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ (𝐴 ↑𝑜 𝑋)) → ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ On)
125	14, 123, 124	syl2anc 575	. . . . . 6 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ On)
126		omcl 7856	. . . . . . 7 ⊢ (((𝐴 ↑𝑜 𝑋) ∈ On ∧ (𝐹‘𝑋) ∈ On) → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) ∈ On)
127	14, 109, 126	syl2anc 575	. . . . . 6 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) ∈ On)
128		oaord 7867	. . . . . 6 ⊢ ((((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ On ∧ (𝐴 ↑𝑜 𝑋) ∈ On ∧ ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) ∈ On) → (((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ (𝐴 ↑𝑜 𝑋) ↔ (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)))) ∈ (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 (𝐴 ↑𝑜 𝑋))))
129	125, 14, 127, 128	syl3anc 1483	. . . . 5 ⊢ (𝜑 → (((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ (𝐴 ↑𝑜 𝑋) ↔ (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)))) ∈ (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 (𝐴 ↑𝑜 𝑋))))
130	123, 129	mpbid 223	. . . 4 ⊢ (𝜑 → (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)))) ∈ (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 (𝐴 ↑𝑜 𝑋)))
131	122, 130	sseldd 3806	. . 3 ⊢ (𝜑 → (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)))) ∈ ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)))
132	107, 131	eqeltrd 2892	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)))
133	56, 132	sseldd 3806	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ (𝐻‘suc (◡𝑂‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   ∨ wo 865   = wceq 1637   ∈ wcel 2157   ≠ wne 2985  ∀wral 3103  ∃wrex 3104  {crab 3107  Vcvv 3398   ∖ cdif 3773   ⊆ wss 3776  ∅c0 4123  ifcif 4286  ∪ cuni 4637   class class class wbr 4851  {copab 4913   ↦ cmpt 4930   E cep 5230   We wwe 5276  ^◡ccnv 5317  dom cdm 5318  Ord word 5942  Oncon0 5943  suc csuc 5945  ⟶wf 6100  –1-1-onto→wf1o 6103  ‘cfv 6104   Isom wiso 6105  (class class class)co 6877   ↦ cmpt2 6879  ωcom 7298   supp csupp 7532  seq_𝜔cseqom 7781   +_𝑜 coa 7796   ·_𝑜 comu 7797   ↑_𝑜 coe 7798   finSupp cfsupp 8517  OrdIsocoi 8656   CNF ccnf 8808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-se 5278  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-isom 6113  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-supp 7533  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-seqom 7782  df-1o 7799  df-2o 7800  df-oadd 7803  df-omul 7804  df-oexp 7805  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fsupp 8518  df-oi 8657  df-cnf 8809

This theorem is referenced by:  cantnflem1  8836