Theorem cantnflem1d 8468

Description: Lemma for cantnf 8473. (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 8464	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
10	9	simp1d 1066	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		onelon 5665	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
12	2, 10, 11	syl2anc 691	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On)
13		oecl 7504	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑𝑜 𝑋) ∈ On)
14	1, 12, 13	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝐴 ↑𝑜 𝑋) ∈ On)
15	3, 1, 2	cantnfs 8446	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
16	6, 15	mpbid 221	. . . . . . . 8 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
17	16	simpld 474	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
18	17, 10	ffvelrnd 6268	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
19		onelon 5665	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑋) ∈ 𝐴) → (𝐺‘𝑋) ∈ On)
20	1, 18, 19	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ On)
21		omcl 7503	. . . . 5 ⊢ (((𝐴 ↑𝑜 𝑋) ∈ On ∧ (𝐺‘𝑋) ∈ On) → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) ∈ On)
22	14, 20, 21	syl2anc 691	. . . 4 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) ∈ On)
23		suppssdm 7195	. . . . . . . . . . . 12 ⊢ (𝐺 supp ∅) ⊆ dom 𝐺
24		fdm 5964	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐵⟶𝐴 → dom 𝐺 = 𝐵)
25	17, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐺 = 𝐵)
26	23, 25	syl5sseq 3616	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
27	2, 26	ssexd 4733	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 supp ∅) ∈ V)
28		cantnflem1.o	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))
29	3, 1, 2, 28, 6	cantnfcl 8447	. . . . . . . . . . 11 ⊢ (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
30	29	simpld 474	. . . . . . . . . 10 ⊢ (𝜑 → E We (𝐺 supp ∅))
31	28	oiiso 8325	. . . . . . . . . 10 ⊢ (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
32	27, 30, 31	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
33		isof1o 6473	. . . . . . . . 9 ⊢ (𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) → 𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅))
34	32, 33	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅))
35		f1ocnv 6062	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) → ◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
36		f1of 6050	. . . . . . . 8 ⊢ (◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂 → ◡𝑂:(𝐺 supp ∅)⟶dom 𝑂)
37	34, 35, 36	3syl 18	. . . . . . 7 ⊢ (𝜑 → ◡𝑂:(𝐺 supp ∅)⟶dom 𝑂)
38	3, 1, 2, 4, 5, 6, 7, 8	cantnflem1a 8465	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))
39	37, 38	ffvelrnd 6268	. . . . . 6 ⊢ (𝜑 → (◡𝑂‘𝑋) ∈ dom 𝑂)
40	29	simprd 478	. . . . . 6 ⊢ (𝜑 → dom 𝑂 ∈ ω)
41		elnn 6967	. . . . . 6 ⊢ (((◡𝑂‘𝑋) ∈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → (◡𝑂‘𝑋) ∈ ω)
42	39, 40, 41	syl2anc 691	. . . . 5 ⊢ (𝜑 → (◡𝑂‘𝑋) ∈ ω)
43		cantnflem1.h	. . . . . . 7 ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝑂‘𝑘)) ·𝑜 (𝐺‘(𝑂‘𝑘))) +𝑜 𝑧)), ∅)
44	43	cantnfvalf 8445	. . . . . 6 ⊢ 𝐻:ω⟶On
45	44	ffvelrni 6266	. . . . 5 ⊢ ((◡𝑂‘𝑋) ∈ ω → (𝐻‘(◡𝑂‘𝑋)) ∈ On)
46	42, 45	syl 17	. . . 4 ⊢ (𝜑 → (𝐻‘(◡𝑂‘𝑋)) ∈ On)
47		oaword1 7519	. . . 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))))
48	22, 46, 47	syl2anc 691	. . 3 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) ⊆ (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) +𝑜 (𝐻‘(◡𝑂‘𝑋))))
49	3, 1, 2, 28, 6, 43	cantnfsuc 8450	. . . . 5 ⊢ ((𝜑 ∧ (◡𝑂‘𝑋) ∈ ω) → (𝐻‘suc (◡𝑂‘𝑋)) = (((𝐴 ↑𝑜 (𝑂‘(◡𝑂‘𝑋))) ·𝑜 (𝐺‘(𝑂‘(◡𝑂‘𝑋)))) +𝑜 (𝐻‘(◡𝑂‘𝑋))))
50	42, 49	mpdan 699	. . . 4 ⊢ (𝜑 → (𝐻‘suc (◡𝑂‘𝑋)) = (((𝐴 ↑𝑜 (𝑂‘(◡𝑂‘𝑋))) ·𝑜 (𝐺‘(𝑂‘(◡𝑂‘𝑋)))) +𝑜 (𝐻‘(◡𝑂‘𝑋))))
51		f1ocnvfv2 6433	. . . . . . . 8 ⊢ ((𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(◡𝑂‘𝑋)) = 𝑋)
52	34, 38, 51	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(◡𝑂‘𝑋)) = 𝑋)
53	52	oveq2d 6565	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑𝑜 (𝑂‘(◡𝑂‘𝑋))) = (𝐴 ↑𝑜 𝑋))
54	52	fveq2d 6107	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑂‘(◡𝑂‘𝑋))) = (𝐺‘𝑋))
55	53, 54	oveq12d 6567	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑𝑜 (𝑂‘(◡𝑂‘𝑋))) ·𝑜 (𝐺‘(𝑂‘(◡𝑂‘𝑋)))) = ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)))
56	55	oveq1d 6564	. . . 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))))
57	50, 56	eqtrd 2644	. . 3 ⊢ (𝜑 → (𝐻‘suc (◡𝑂‘𝑋)) = (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) +𝑜 (𝐻‘(◡𝑂‘𝑋))))
58	48, 57	sseqtr4d 3605	. 2 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)) ⊆ (𝐻‘suc (◡𝑂‘𝑋)))
59		onss 6882	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
60	2, 59	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ On)
61	60	sselda 3568	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
62	12	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ On)
63		onsseleq 5682	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥 ⊆ 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
64	61, 62, 63	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
65		orcom 401	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋) ↔ (𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋))
66	64, 65	syl6bb 275	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝑋 ↔ (𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋)))
67	66	ifbid 4058	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅) = if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅))
68	67	mpteq2dva 4672	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅)) = (𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅)))
69	68	fveq2d 6107	. . . 4 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) = ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅))))
70	3, 1, 2	cantnfs 8446	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
71	5, 70	mpbid 221	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))
72	71	simpld 474	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
73	72	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ 𝐴)
74		ne0i 3880	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑋) ∈ 𝐴 → 𝐴 ≠ ∅)
75	18, 74	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≠ ∅)
76		on0eln0 5697	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
77	1, 76	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
78	75, 77	mpbird 246	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ 𝐴)
79	78	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ 𝐴)
80	73, 79	ifcld 4081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) ∈ 𝐴)
81		eqid 2610	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) = (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))
82	80, 81	fmptd 6292	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)):𝐵⟶𝐴)
83		0ex 4718	. . . . . . . . 9 ⊢ ∅ ∈ V
84	83	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ V)
85	71	simprd 478	. . . . . . . 8 ⊢ (𝜑 → 𝐹 finSupp ∅)
86	72, 2, 84, 85	fsuppmptif 8188	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) finSupp ∅)
87	3, 1, 2	cantnfs 8446	. . . . . . 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 ∅)))
88	82, 86, 87	mpbir2and 959	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) ∈ 𝑆)
89	72, 10	ffvelrnd 6268	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐴)
90		eldifn 3695	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∖ 𝑋) → ¬ 𝑦 ∈ 𝑋)
91	90	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ 𝑋)) → ¬ 𝑦 ∈ 𝑋)
92	91	iffalsed 4047	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ 𝑋)) → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) = ∅)
93	92, 2	suppss2 7216	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) supp ∅) ⊆ 𝑋)
94		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
95	94	adantl 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
96	95	ifeq1da 4066	. . . . . . . . 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)))
97		eleq1 2676	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋))
98		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
99	97, 98	ifbieq1d 4059	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) = if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
100		fvex 6113	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑥) ∈ V
101	100, 83	ifex 4106	. . . . . . . . . . 11 ⊢ if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅) ∈ V
102	99, 81, 101	fvmpt 6191	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥) = if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
103	102	ifeq2d 4055	. . . . . . . . 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), ∅)))
104	96, 103	eqtr3d 2646	. . . . . . . 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), ∅)))
105		ifor 4085	. . . . . . . 8 ⊢ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅) = if(𝑥 = 𝑋, (𝐹‘𝑥), if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
106	104, 105	syl6reqr 2663	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅) = if(𝑥 = 𝑋, (𝐹‘𝑋), ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥)))
107	106	mpteq2ia 4668	. . . . . 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)))
108	3, 1, 2, 88, 10, 89, 93, 107	cantnfp1 8461	. . . . 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), ∅))))))
109	108	simprd 478	. . . 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), ∅)))))
110	69, 109	eqtrd 2644	. . 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), ∅)))))
111		onelon 5665	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐹‘𝑋) ∈ 𝐴) → (𝐹‘𝑋) ∈ On)
112	1, 89, 111	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ On)
113		omsuc 7493	. . . . . 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)))
114	14, 112, 113	syl2anc 691	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 suc (𝐹‘𝑋)) = (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 (𝐴 ↑𝑜 𝑋)))
115		eloni 5650	. . . . . . . 8 ⊢ ((𝐺‘𝑋) ∈ On → Ord (𝐺‘𝑋))
116	20, 115	syl 17	. . . . . . 7 ⊢ (𝜑 → Ord (𝐺‘𝑋))
117	9	simp2d 1067	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐺‘𝑋))
118		ordsucss 6910	. . . . . . 7 ⊢ (Ord (𝐺‘𝑋) → ((𝐹‘𝑋) ∈ (𝐺‘𝑋) → suc (𝐹‘𝑋) ⊆ (𝐺‘𝑋)))
119	116, 117, 118	sylc 63	. . . . . 6 ⊢ (𝜑 → suc (𝐹‘𝑋) ⊆ (𝐺‘𝑋))
120		suceloni 6905	. . . . . . . 8 ⊢ ((𝐹‘𝑋) ∈ On → suc (𝐹‘𝑋) ∈ On)
121	112, 120	syl 17	. . . . . . 7 ⊢ (𝜑 → suc (𝐹‘𝑋) ∈ On)
122		omwordi 7538	. . . . . . 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))))
123	121, 20, 14, 122	syl3anc 1318	. . . . . 6 ⊢ (𝜑 → (suc (𝐹‘𝑋) ⊆ (𝐺‘𝑋) → ((𝐴 ↑𝑜 𝑋) ·𝑜 suc (𝐹‘𝑋)) ⊆ ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋))))
124	119, 123	mpd 15	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 suc (𝐹‘𝑋)) ⊆ ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)))
125	114, 124	eqsstr3d 3603	. . . 4 ⊢ (𝜑 → (((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) +𝑜 (𝐴 ↑𝑜 𝑋)) ⊆ ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)))
126	3, 1, 2, 88, 78, 12, 93	cantnflt2 8453	. . . . 5 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ (𝐴 ↑𝑜 𝑋))
127		onelon 5665	. . . . . . 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)
128	14, 126, 127	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ On)
129		omcl 7503	. . . . . . 7 ⊢ (((𝐴 ↑𝑜 𝑋) ∈ On ∧ (𝐹‘𝑋) ∈ On) → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) ∈ On)
130	14, 112, 129	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐹‘𝑋)) ∈ On)
131		oaord 7514	. . . . . 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))))
132	128, 14, 130, 131	syl3anc 1318	. . . . 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))))
133	126, 132	mpbid 221	. . . 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)))
134	125, 133	sseldd 3569	. . 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)))
135	110, 134	eqeltrd 2688	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ ((𝐴 ↑𝑜 𝑋) ·𝑜 (𝐺‘𝑋)))
136	58, 135	sseldd 3569	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ (𝐻‘suc (◡𝑂‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ifcif 4036  ∪ cuni 4372   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   E cep 4947   We wwe 4996  ^◡ccnv 5037  dom cdm 5038  Ord word 5639  Oncon0 5640  suc csuc 5642  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   supp csupp 7182  seq_𝜔cseqom 7429   +_𝑜 coa 7444   ·_𝑜 comu 7445   ↑_𝑜 coe 7446   finSupp cfsupp 8158  OrdIsocoi 8297   CNF ccnf 8441

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-fal 1481  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-rmo 2904  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-int 4411  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-se 4998  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-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-cnf 8442

This theorem is referenced by:  cantnflem1  8469