cantnflem1d - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cantnflem1d		Structured version Visualization version GIF version

Theorem cantnflem1d 9376

Description: Lemma for cantnf 9381. (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 = seq_ω((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 9372	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
10	9	simp1d 1140	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		onelon 6276	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
12	2, 10, 11	syl2anc 583	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On)
13		oecl 8329	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑_o 𝑋) ∈ On)
14	1, 12, 13	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐴 ↑_o 𝑋) ∈ On)
15	3, 1, 2	cantnfs 9354	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
16	6, 15	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
17	16	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
18	17, 10	ffvelrnd 6944	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
19		onelon 6276	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑋) ∈ 𝐴) → (𝐺‘𝑋) ∈ On)
20	1, 18, 19	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ On)
21		omcl 8328	. . . . 5 ⊢ (((𝐴 ↑_o 𝑋) ∈ On ∧ (𝐺‘𝑋) ∈ On) → ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)) ∈ On)
22	14, 20, 21	syl2anc 583	. . . 4 ⊢ (𝜑 → ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)) ∈ On)
23		ovexd 7290	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 supp ∅) ∈ V)
24		cantnflem1.o	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))
25	3, 1, 2, 24, 6	cantnfcl 9355	. . . . . . . . . . 11 ⊢ (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
26	25	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → E We (𝐺 supp ∅))
27	24	oiiso 9226	. . . . . . . . . 10 ⊢ (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
28	23, 26, 27	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
29		isof1o 7174	. . . . . . . . 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 6712	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) → ^◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
32		f1of 6700	. . . . . . . 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 9373	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))
35	33, 34	ffvelrnd 6944	. . . . . 6 ⊢ (𝜑 → (^◡𝑂‘𝑋) ∈ dom 𝑂)
36	25	simprd 495	. . . . . 6 ⊢ (𝜑 → dom 𝑂 ∈ ω)
37		elnn 7698	. . . . . 6 ⊢ (((^◡𝑂‘𝑋) ∈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → (^◡𝑂‘𝑋) ∈ ω)
38	35, 36, 37	syl2anc 583	. . . . 5 ⊢ (𝜑 → (^◡𝑂‘𝑋) ∈ ω)
39		cantnflem1.h	. . . . . . 7 ⊢ 𝐻 = seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_o (𝑂‘𝑘)) ·_o (𝐺‘(𝑂‘𝑘))) +_o 𝑧)), ∅)
40	39	cantnfvalf 9353	. . . . . 6 ⊢ 𝐻:ω⟶On
41	40	ffvelrni 6942	. . . . 5 ⊢ ((^◡𝑂‘𝑋) ∈ ω → (𝐻‘(^◡𝑂‘𝑋)) ∈ On)
42	38, 41	syl 17	. . . 4 ⊢ (𝜑 → (𝐻‘(^◡𝑂‘𝑋)) ∈ On)
43		oaword1 8345	. . . 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 583	. . 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 9358	. . . . 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 683	. . . 4 ⊢ (𝜑 → (𝐻‘suc (^◡𝑂‘𝑋)) = (((𝐴 ↑_o (𝑂‘(^◡𝑂‘𝑋))) ·_o (𝐺‘(𝑂‘(^◡𝑂‘𝑋)))) +_o (𝐻‘(^◡𝑂‘𝑋))))
47		f1ocnvfv2 7130	. . . . . . . 8 ⊢ ((𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(^◡𝑂‘𝑋)) = 𝑋)
48	30, 34, 47	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(^◡𝑂‘𝑋)) = 𝑋)
49	48	oveq2d 7271	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑_o (𝑂‘(^◡𝑂‘𝑋))) = (𝐴 ↑_o 𝑋))
50	48	fveq2d 6760	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑂‘(^◡𝑂‘𝑋))) = (𝐺‘𝑋))
51	49, 50	oveq12d 7273	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑_o (𝑂‘(^◡𝑂‘𝑋))) ·_o (𝐺‘(𝑂‘(^◡𝑂‘𝑋)))) = ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)))
52	51	oveq1d 7270	. . . 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 2778	. . 3 ⊢ (𝜑 → (𝐻‘suc (^◡𝑂‘𝑋)) = (((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)) +_o (𝐻‘(^◡𝑂‘𝑋))))
54	44, 53	sseqtrrd 3958	. 2 ⊢ (𝜑 → ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)) ⊆ (𝐻‘suc (^◡𝑂‘𝑋)))
55		onss 7611	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
56	2, 55	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ On)
57	56	sselda 3917	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
58	12	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ On)
59		onsseleq 6292	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥 ⊆ 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
60	57, 58, 59	syl2anc 583	. . . . . . . 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	bitrdi 286	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝑋 ↔ (𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋)))
63	62	ifbid 4479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅) = if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅))
64	63	mpteq2dva 5170	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅)) = (𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅)))
65	64	fveq2d 6760	. . . 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 9354	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
67	5, 66	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))
68	67	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
69	68	ffvelrnda 6943	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ 𝐴)
70	18	ne0d 4266	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≠ ∅)
71		on0eln0 6306	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
72	1, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
73	70, 72	mpbird 256	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ 𝐴)
74	73	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ 𝐴)
75	69, 74	ifcld 4502	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) ∈ 𝐴)
76	75	fmpttd 6971	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)):𝐵⟶𝐴)
77		0ex 5226	. . . . . . . . 9 ⊢ ∅ ∈ V
78	77	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ V)
79	67	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝐹 finSupp ∅)
80	68, 2, 78, 79	fsuppmptif 9088	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) finSupp ∅)
81	3, 1, 2	cantnfs 9354	. . . . . . 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 709	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) ∈ 𝑆)
83	68, 10	ffvelrnd 6944	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐴)
84		eldifn 4058	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∖ 𝑋) → ¬ 𝑦 ∈ 𝑋)
85	84	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ 𝑋)) → ¬ 𝑦 ∈ 𝑋)
86	85	iffalsed 4467	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ 𝑋)) → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) = ∅)
87	86, 2	suppss2 7987	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) supp ∅) ⊆ 𝑋)
88		ifor 4510	. . . . . . . 8 ⊢ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅) = if(𝑥 = 𝑋, (𝐹‘𝑥), if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
89		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
90	89	adantl 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
91	90	ifeq1da 4487	. . . . . . . . 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)))
92		eleq1w 2821	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋))
93		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
94	92, 93	ifbieq1d 4480	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) = if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
95		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) = (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))
96		fvex 6769	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑥) ∈ V
97	96, 77	ifex 4506	. . . . . . . . . . 11 ⊢ if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅) ∈ V
98	94, 95, 97	fvmpt 6857	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥) = if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
99	98	ifeq2d 4476	. . . . . . . . 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), ∅)))
100	91, 99	eqtr3d 2780	. . . . . . . 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), ∅)))
101	88, 100	eqtr4id 2798	. . . . . . 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 5173	. . . . . 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 9369	. . . . 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 495	. . . 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 2778	. . 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 6276	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐹‘𝑋) ∈ 𝐴) → (𝐹‘𝑋) ∈ On)
107	1, 83, 106	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ On)
108		omsuc 8318	. . . . . 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 583	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑_o 𝑋) ·_o suc (𝐹‘𝑋)) = (((𝐴 ↑_o 𝑋) ·_o (𝐹‘𝑋)) +_o (𝐴 ↑_o 𝑋)))
110		eloni 6261	. . . . . . . 8 ⊢ ((𝐺‘𝑋) ∈ On → Ord (𝐺‘𝑋))
111	20, 110	syl 17	. . . . . . 7 ⊢ (𝜑 → Ord (𝐺‘𝑋))
112	9	simp2d 1141	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐺‘𝑋))
113		ordsucss 7640	. . . . . . 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 7635	. . . . . . . 8 ⊢ ((𝐹‘𝑋) ∈ On → suc (𝐹‘𝑋) ∈ On)
116	107, 115	syl 17	. . . . . . 7 ⊢ (𝜑 → suc (𝐹‘𝑋) ∈ On)
117		omwordi 8364	. . . . . . 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 1369	. . . . . 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 3956	. . . 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 9361	. . . . 5 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ (𝐴 ↑_o 𝑋))
122		onelon 6276	. . . . . . 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 583	. . . . . 6 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ On)
124		omcl 8328	. . . . . . 7 ⊢ (((𝐴 ↑_o 𝑋) ∈ On ∧ (𝐹‘𝑋) ∈ On) → ((𝐴 ↑_o 𝑋) ·_o (𝐹‘𝑋)) ∈ On)
125	14, 107, 124	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ((𝐴 ↑_o 𝑋) ·_o (𝐹‘𝑋)) ∈ On)
126		oaord 8340	. . . . . 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 1369	. . . . 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 231	. . . 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 3918	. . 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 2839	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)))
131	54, 130	sseldd 3918	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ (𝐻‘suc (^◡𝑂‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 Vcvv 3422 ∖ cdif 3880 ⊆ wss 3883 ∅c0 4253 ifcif 4456 ∪ cuni 4836 class class class wbr 5070 {copab 5132 ↦ cmpt 5153 E cep 5485 We wwe 5534 ^◡ccnv 5579 dom cdm 5580 Ord word 6250 Oncon0 6251 suc csuc 6253 ⟶wf 6414 –1-1-onto→wf1o 6417 ‘cfv 6418 Isom wiso 6419 (class class class)co 7255 ∈ cmpo 7257 ωcom 7687 supp csupp 7948 seq_ωcseqom 8248 +_o coa 8264 ·_o comu 8265 ↑_o coe 8266 finSupp cfsupp 9058 OrdIsocoi 9198 CNF ccnf 9349

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-seqom 8249 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-oexp 8273 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-oi 9199 df-cnf 9350

This theorem is referenced by: cantnflem1 9377

W3C validator