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Theorem cantnflem1d 9641

Description: Lemma for cantnf 9646. (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 9637	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
10	9	simp1d 1142	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		onelon 6357	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
12	2, 10, 11	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On)
13		oecl 8501	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑_o 𝑋) ∈ On)
14	1, 12, 13	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐴 ↑_o 𝑋) ∈ On)
15	3, 1, 2	cantnfs 9619	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
16	6, 15	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
17	16	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
18	17, 10	ffvelcdmd 7057	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
19		onelon 6357	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑋) ∈ 𝐴) → (𝐺‘𝑋) ∈ On)
20	1, 18, 19	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ On)
21		omcl 8500	. . . . 5 ⊢ (((𝐴 ↑_o 𝑋) ∈ On ∧ (𝐺‘𝑋) ∈ On) → ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)) ∈ On)
22	14, 20, 21	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)) ∈ On)
23		ovexd 7422	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 supp ∅) ∈ V)
24		cantnflem1.o	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))
25	3, 1, 2, 24, 6	cantnfcl 9620	. . . . . . . . . . 11 ⊢ (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
26	25	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → E We (𝐺 supp ∅))
27	24	oiiso 9490	. . . . . . . . . 10 ⊢ (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
28	23, 26, 27	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
29		isof1o 7298	. . . . . . . . 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 6812	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) → ^◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
32		f1of 6800	. . . . . . . 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 9638	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))
35	33, 34	ffvelcdmd 7057	. . . . . 6 ⊢ (𝜑 → (^◡𝑂‘𝑋) ∈ dom 𝑂)
36	25	simprd 495	. . . . . 6 ⊢ (𝜑 → dom 𝑂 ∈ ω)
37		elnn 7853	. . . . . 6 ⊢ (((^◡𝑂‘𝑋) ∈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → (^◡𝑂‘𝑋) ∈ ω)
38	35, 36, 37	syl2anc 584	. . . . 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 9618	. . . . . 6 ⊢ 𝐻:ω⟶On
41	40	ffvelcdmi 7055	. . . . 5 ⊢ ((^◡𝑂‘𝑋) ∈ ω → (𝐻‘(^◡𝑂‘𝑋)) ∈ On)
42	38, 41	syl 17	. . . 4 ⊢ (𝜑 → (𝐻‘(^◡𝑂‘𝑋)) ∈ On)
43		oaword1 8516	. . . 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 584	. . 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 9623	. . . . 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 687	. . . 4 ⊢ (𝜑 → (𝐻‘suc (^◡𝑂‘𝑋)) = (((𝐴 ↑_o (𝑂‘(^◡𝑂‘𝑋))) ·_o (𝐺‘(𝑂‘(^◡𝑂‘𝑋)))) +_o (𝐻‘(^◡𝑂‘𝑋))))
47		f1ocnvfv2 7252	. . . . . . . 8 ⊢ ((𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(^◡𝑂‘𝑋)) = 𝑋)
48	30, 34, 47	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(^◡𝑂‘𝑋)) = 𝑋)
49	48	oveq2d 7403	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑_o (𝑂‘(^◡𝑂‘𝑋))) = (𝐴 ↑_o 𝑋))
50	48	fveq2d 6862	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑂‘(^◡𝑂‘𝑋))) = (𝐺‘𝑋))
51	49, 50	oveq12d 7405	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑_o (𝑂‘(^◡𝑂‘𝑋))) ·_o (𝐺‘(𝑂‘(^◡𝑂‘𝑋)))) = ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)))
52	51	oveq1d 7402	. . . 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 2764	. . 3 ⊢ (𝜑 → (𝐻‘suc (^◡𝑂‘𝑋)) = (((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)) +_o (𝐻‘(^◡𝑂‘𝑋))))
54	44, 53	sseqtrrd 3984	. 2 ⊢ (𝜑 → ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)) ⊆ (𝐻‘suc (^◡𝑂‘𝑋)))
55		onss 7761	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
56	2, 55	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ On)
57	56	sselda 3946	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
58	12	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ On)
59		onsseleq 6373	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥 ⊆ 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
60	57, 58, 59	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
61		orcom 870	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋) ↔ (𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋))
62	60, 61	bitrdi 287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝑋 ↔ (𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋)))
63	62	ifbid 4512	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅) = if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅))
64	63	mpteq2dva 5200	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅)) = (𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅)))
65	64	fveq2d 6862	. . . 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 9619	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
67	5, 66	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))
68	67	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
69	68	ffvelcdmda 7056	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ 𝐴)
70	18	ne0d 4305	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≠ ∅)
71		on0eln0 6389	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
72	1, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
73	70, 72	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ 𝐴)
74	73	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ 𝐴)
75	69, 74	ifcld 4535	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) ∈ 𝐴)
76	75	fmpttd 7087	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)):𝐵⟶𝐴)
77		0ex 5262	. . . . . . . . 9 ⊢ ∅ ∈ V
78	77	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ V)
79	67	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝐹 finSupp ∅)
80	68, 2, 78, 79	fsuppmptif 9350	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) finSupp ∅)
81	3, 1, 2	cantnfs 9619	. . . . . . 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 713	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) ∈ 𝑆)
83	68, 10	ffvelcdmd 7057	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐴)
84		eldifn 4095	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∖ 𝑋) → ¬ 𝑦 ∈ 𝑋)
85	84	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ 𝑋)) → ¬ 𝑦 ∈ 𝑋)
86	85	iffalsed 4499	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ 𝑋)) → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) = ∅)
87	86, 2	suppss2 8179	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) supp ∅) ⊆ 𝑋)
88		ifor 4543	. . . . . . . 8 ⊢ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅) = if(𝑥 = 𝑋, (𝐹‘𝑥), if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
89		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
90	89	adantl 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
91	90	ifeq1da 4520	. . . . . . . . 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 2811	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋))
93		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
94	92, 93	ifbieq1d 4513	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) = if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
95		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) = (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))
96		fvex 6871	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑥) ∈ V
97	96, 77	ifex 4539	. . . . . . . . . . 11 ⊢ if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅) ∈ V
98	94, 95, 97	fvmpt 6968	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥) = if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
99	98	ifeq2d 4509	. . . . . . . . 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 2766	. . . . . . . 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 2783	. . . . . . 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 5202	. . . . . 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 9634	. . . . 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 2764	. . 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 6357	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐹‘𝑋) ∈ 𝐴) → (𝐹‘𝑋) ∈ On)
107	1, 83, 106	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ On)
108		omsuc 8490	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑_o 𝑋) ·_o suc (𝐹‘𝑋)) = (((𝐴 ↑_o 𝑋) ·_o (𝐹‘𝑋)) +_o (𝐴 ↑_o 𝑋)))
110		eloni 6342	. . . . . . . 8 ⊢ ((𝐺‘𝑋) ∈ On → Ord (𝐺‘𝑋))
111	20, 110	syl 17	. . . . . . 7 ⊢ (𝜑 → Ord (𝐺‘𝑋))
112	9	simp2d 1143	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐺‘𝑋))
113		ordsucss 7793	. . . . . . 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		onsuc 7787	. . . . . . . 8 ⊢ ((𝐹‘𝑋) ∈ On → suc (𝐹‘𝑋) ∈ On)
116	107, 115	syl 17	. . . . . . 7 ⊢ (𝜑 → suc (𝐹‘𝑋) ∈ On)
117		omwordi 8535	. . . . . . 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 1373	. . . . . 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 3982	. . . 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 9626	. . . . 5 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ (𝐴 ↑_o 𝑋))
122		onelon 6357	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ On)
124		omcl 8500	. . . . . . 7 ⊢ (((𝐴 ↑_o 𝑋) ∈ On ∧ (𝐹‘𝑋) ∈ On) → ((𝐴 ↑_o 𝑋) ·_o (𝐹‘𝑋)) ∈ On)
125	14, 107, 124	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((𝐴 ↑_o 𝑋) ·_o (𝐹‘𝑋)) ∈ On)
126		oaord 8511	. . . . . 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 1373	. . . . 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 232	. . . 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 3947	. . 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 2828	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)))
131	54, 130	sseldd 3947	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 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3405 Vcvv 3447 ∖ cdif 3911 ⊆ wss 3914 ∅c0 4296 ifcif 4488 ∪ cuni 4871 class class class wbr 5107 {copab 5169 ↦ cmpt 5188 E cep 5537 We wwe 5590 ^◡ccnv 5637 dom cdm 5638 Ord word 6331 Oncon0 6332 suc csuc 6334 ⟶wf 6507 –1-1-onto→wf1o 6510 ‘cfv 6511 Isom wiso 6512 (class class class)co 7387 ∈ cmpo 7389 ωcom 7842 supp csupp 8139 seq_ωcseqom 8415 +_o coa 8431 ·_o comu 8432 ↑_o coe 8433 finSupp cfsupp 9312 OrdIsocoi 9462 CNF ccnf 9614

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-seqom 8416 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-oexp 8440 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-oi 9463 df-cnf 9615

This theorem is referenced by: cantnflem1 9642

W3C validator