Theorem cantnflem1d 9145

Description: Lemma for cantnf 9150. (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 9141	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
10	9	simp1d 1138	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		onelon 6211	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
12	2, 10, 11	syl2anc 586	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On)
13		oecl 8156	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑o 𝑋) ∈ On)
14	1, 12, 13	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐴 ↑o 𝑋) ∈ On)
15	3, 1, 2	cantnfs 9123	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
16	6, 15	mpbid 234	. . . . . . . 8 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
17	16	simpld 497	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
18	17, 10	ffvelrnd 6847	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
19		onelon 6211	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑋) ∈ 𝐴) → (𝐺‘𝑋) ∈ On)
20	1, 18, 19	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ On)
21		omcl 8155	. . . . 5 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ (𝐺‘𝑋) ∈ On) → ((𝐴 ↑o 𝑋) ·o (𝐺‘𝑋)) ∈ On)
22	14, 20, 21	syl2anc 586	. . . 4 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o (𝐺‘𝑋)) ∈ On)
23		ovexd 7185	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 supp ∅) ∈ V)
24		cantnflem1.o	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))
25	3, 1, 2, 24, 6	cantnfcl 9124	. . . . . . . . . . 11 ⊢ (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
26	25	simpld 497	. . . . . . . . . 10 ⊢ (𝜑 → E We (𝐺 supp ∅))
27	24	oiiso 8995	. . . . . . . . . 10 ⊢ (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
28	23, 26, 27	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
29		isof1o 7070	. . . . . . . . 9 ⊢ (𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) → 𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅))
30	28, 29	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅))
31		f1ocnv 6622	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) → ◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
32		f1of 6610	. . . . . . . 8 ⊢ (◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂 → ◡𝑂:(𝐺 supp ∅)⟶dom 𝑂)
33	30, 31, 32	3syl 18	. . . . . . 7 ⊢ (𝜑 → ◡𝑂:(𝐺 supp ∅)⟶dom 𝑂)
34	3, 1, 2, 4, 5, 6, 7, 8	cantnflem1a 9142	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))
35	33, 34	ffvelrnd 6847	. . . . . 6 ⊢ (𝜑 → (◡𝑂‘𝑋) ∈ dom 𝑂)
36	25	simprd 498	. . . . . 6 ⊢ (𝜑 → dom 𝑂 ∈ ω)
37		elnn 7584	. . . . . 6 ⊢ (((◡𝑂‘𝑋) ∈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → (◡𝑂‘𝑋) ∈ ω)
38	35, 36, 37	syl2anc 586	. . . . 5 ⊢ (𝜑 → (◡𝑂‘𝑋) ∈ ω)
39		cantnflem1.h	. . . . . . 7 ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐺‘(𝑂‘𝑘))) +o 𝑧)), ∅)
40	39	cantnfvalf 9122	. . . . . 6 ⊢ 𝐻:ω⟶On
41	40	ffvelrni 6845	. . . . 5 ⊢ ((◡𝑂‘𝑋) ∈ ω → (𝐻‘(◡𝑂‘𝑋)) ∈ On)
42	38, 41	syl 17	. . . 4 ⊢ (𝜑 → (𝐻‘(◡𝑂‘𝑋)) ∈ On)
43		oaword1 8172	. . . 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))))
44	22, 42, 43	syl2anc 586	. . 3 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o (𝐺‘𝑋)) ⊆ (((𝐴 ↑o 𝑋) ·o (𝐺‘𝑋)) +o (𝐻‘(◡𝑂‘𝑋))))
45	3, 1, 2, 24, 6, 39	cantnfsuc 9127	. . . . 5 ⊢ ((𝜑 ∧ (◡𝑂‘𝑋) ∈ ω) → (𝐻‘suc (◡𝑂‘𝑋)) = (((𝐴 ↑o (𝑂‘(◡𝑂‘𝑋))) ·o (𝐺‘(𝑂‘(◡𝑂‘𝑋)))) +o (𝐻‘(◡𝑂‘𝑋))))
46	38, 45	mpdan 685	. . . 4 ⊢ (𝜑 → (𝐻‘suc (◡𝑂‘𝑋)) = (((𝐴 ↑o (𝑂‘(◡𝑂‘𝑋))) ·o (𝐺‘(𝑂‘(◡𝑂‘𝑋)))) +o (𝐻‘(◡𝑂‘𝑋))))
47		f1ocnvfv2 7028	. . . . . . . 8 ⊢ ((𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(◡𝑂‘𝑋)) = 𝑋)
48	30, 34, 47	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(◡𝑂‘𝑋)) = 𝑋)
49	48	oveq2d 7166	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑o (𝑂‘(◡𝑂‘𝑋))) = (𝐴 ↑o 𝑋))
50	48	fveq2d 6669	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑂‘(◡𝑂‘𝑋))) = (𝐺‘𝑋))
51	49, 50	oveq12d 7168	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑o (𝑂‘(◡𝑂‘𝑋))) ·o (𝐺‘(𝑂‘(◡𝑂‘𝑋)))) = ((𝐴 ↑o 𝑋) ·o (𝐺‘𝑋)))
52	51	oveq1d 7165	. . . 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))))
53	46, 52	eqtrd 2856	. . 3 ⊢ (𝜑 → (𝐻‘suc (◡𝑂‘𝑋)) = (((𝐴 ↑o 𝑋) ·o (𝐺‘𝑋)) +o (𝐻‘(◡𝑂‘𝑋))))
54	44, 53	sseqtrrd 4008	. 2 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o (𝐺‘𝑋)) ⊆ (𝐻‘suc (◡𝑂‘𝑋)))
55		onss 7499	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
56	2, 55	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ On)
57	56	sselda 3967	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
58	12	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ On)
59		onsseleq 6227	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥 ⊆ 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
60	57, 58, 59	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
61		orcom 866	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋) ↔ (𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋))
62	60, 61	syl6bb 289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝑋 ↔ (𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋)))
63	62	ifbid 4489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅) = if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅))
64	63	mpteq2dva 5154	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅)) = (𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅)))
65	64	fveq2d 6669	. . . 4 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) = ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅))))
66	3, 1, 2	cantnfs 9123	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
67	5, 66	mpbid 234	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))
68	67	simpld 497	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
69	68	ffvelrnda 6846	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ 𝐴)
70	18	ne0d 4301	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≠ ∅)
71		on0eln0 6241	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
72	1, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
73	70, 72	mpbird 259	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ 𝐴)
74	73	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ 𝐴)
75	69, 74	ifcld 4512	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) ∈ 𝐴)
76	75	fmpttd 6874	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)):𝐵⟶𝐴)
77		0ex 5204	. . . . . . . . 9 ⊢ ∅ ∈ V
78	77	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ V)
79	67	simprd 498	. . . . . . . 8 ⊢ (𝜑 → 𝐹 finSupp ∅)
80	68, 2, 78, 79	fsuppmptif 8857	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) finSupp ∅)
81	3, 1, 2	cantnfs 9123	. . . . . . 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 ∅)))
82	76, 80, 81	mpbir2and 711	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) ∈ 𝑆)
83	68, 10	ffvelrnd 6847	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐴)
84		eldifn 4104	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∖ 𝑋) → ¬ 𝑦 ∈ 𝑋)
85	84	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ 𝑋)) → ¬ 𝑦 ∈ 𝑋)
86	85	iffalsed 4478	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ 𝑋)) → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) = ∅)
87	86, 2	suppss2 7858	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) supp ∅) ⊆ 𝑋)
88		fveq2 6665	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
89	88	adantl 484	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
90	89	ifeq1da 4497	. . . . . . . . 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)))
91		eleq1w 2895	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋))
92		fveq2 6665	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
93	91, 92	ifbieq1d 4490	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) = if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
94		eqid 2821	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) = (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))
95		fvex 6678	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑥) ∈ V
96	95, 77	ifex 4515	. . . . . . . . . . 11 ⊢ if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅) ∈ V
97	93, 94, 96	fvmpt 6763	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥) = if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
98	97	ifeq2d 4486	. . . . . . . . 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), ∅)))
99	90, 98	eqtr3d 2858	. . . . . . . 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), ∅)))
100		ifor 4519	. . . . . . . 8 ⊢ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅) = if(𝑥 = 𝑋, (𝐹‘𝑥), if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
101	99, 100	syl6reqr 2875	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅) = if(𝑥 = 𝑋, (𝐹‘𝑋), ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥)))
102	101	mpteq2ia 5150	. . . . . 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)))
103	3, 1, 2, 82, 10, 83, 87, 102	cantnfp1 9138	. . . . 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), ∅))))))
104	103	simprd 498	. . . 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), ∅)))))
105	65, 104	eqtrd 2856	. . 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), ∅)))))
106		onelon 6211	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐹‘𝑋) ∈ 𝐴) → (𝐹‘𝑋) ∈ On)
107	1, 83, 106	syl2anc 586	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ On)
108		omsuc 8145	. . . . . 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)))
109	14, 107, 108	syl2anc 586	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o suc (𝐹‘𝑋)) = (((𝐴 ↑o 𝑋) ·o (𝐹‘𝑋)) +o (𝐴 ↑o 𝑋)))
110		eloni 6196	. . . . . . . 8 ⊢ ((𝐺‘𝑋) ∈ On → Ord (𝐺‘𝑋))
111	20, 110	syl 17	. . . . . . 7 ⊢ (𝜑 → Ord (𝐺‘𝑋))
112	9	simp2d 1139	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐺‘𝑋))
113		ordsucss 7527	. . . . . . 7 ⊢ (Ord (𝐺‘𝑋) → ((𝐹‘𝑋) ∈ (𝐺‘𝑋) → suc (𝐹‘𝑋) ⊆ (𝐺‘𝑋)))
114	111, 112, 113	sylc 65	. . . . . 6 ⊢ (𝜑 → suc (𝐹‘𝑋) ⊆ (𝐺‘𝑋))
115		suceloni 7522	. . . . . . . 8 ⊢ ((𝐹‘𝑋) ∈ On → suc (𝐹‘𝑋) ∈ On)
116	107, 115	syl 17	. . . . . . 7 ⊢ (𝜑 → suc (𝐹‘𝑋) ∈ On)
117		omwordi 8191	. . . . . . 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))))
118	116, 20, 14, 117	syl3anc 1367	. . . . . 6 ⊢ (𝜑 → (suc (𝐹‘𝑋) ⊆ (𝐺‘𝑋) → ((𝐴 ↑o 𝑋) ·o suc (𝐹‘𝑋)) ⊆ ((𝐴 ↑o 𝑋) ·o (𝐺‘𝑋))))
119	114, 118	mpd 15	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o suc (𝐹‘𝑋)) ⊆ ((𝐴 ↑o 𝑋) ·o (𝐺‘𝑋)))
120	109, 119	eqsstrrd 4006	. . . 4 ⊢ (𝜑 → (((𝐴 ↑o 𝑋) ·o (𝐹‘𝑋)) +o (𝐴 ↑o 𝑋)) ⊆ ((𝐴 ↑o 𝑋) ·o (𝐺‘𝑋)))
121	3, 1, 2, 82, 73, 12, 87	cantnflt2 9130	. . . . 5 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ (𝐴 ↑o 𝑋))
122		onelon 6211	. . . . . . 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)
123	14, 121, 122	syl2anc 586	. . . . . 6 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ On)
124		omcl 8155	. . . . . . 7 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ (𝐹‘𝑋) ∈ On) → ((𝐴 ↑o 𝑋) ·o (𝐹‘𝑋)) ∈ On)
125	14, 107, 124	syl2anc 586	. . . . . 6 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o (𝐹‘𝑋)) ∈ On)
126		oaord 8167	. . . . . 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))))
127	123, 14, 125, 126	syl3anc 1367	. . . . 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))))
128	121, 127	mpbid 234	. . . 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)))
129	120, 128	sseldd 3968	. . 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)))
130	105, 129	eqeltrd 2913	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ ((𝐴 ↑o 𝑋) ·o (𝐺‘𝑋)))
131	54, 130	sseldd 3968	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ (𝐻‘suc (◡𝑂‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3495   ∖ cdif 3933   ⊆ wss 3936  ∅c0 4291  ifcif 4467  ∪ cuni 4832   class class class wbr 5059  {copab 5121   ↦ cmpt 5139   E cep 5459   We wwe 5508  ^◡ccnv 5549  dom cdm 5550  Ord word 6185  Oncon0 6186  suc csuc 6188  ⟶wf 6346  –1-1-onto→wf1o 6349  ‘cfv 6350   Isom wiso 6351  (class class class)co 7150   ∈ cmpo 7152  ωcom 7574   supp csupp 7824  seq_ωcseqom 8077   +_o coa 8093   ·_o comu 8094   ↑_o coe 8095   finSupp cfsupp 8827  OrdIsocoi 8967   CNF ccnf 9118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-seqom 8078  df-1o 8096  df-2o 8097  df-oadd 8100  df-omul 8101  df-oexp 8102  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-oi 8968  df-cnf 9119

This theorem is referenced by:  cantnflem1  9146