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 9682

Description: Lemma for cantnf 9687. (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 9678	. . . . . . . 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 6389	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
12	2, 10, 11	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On)
13		oecl 8536	. . . . . 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 9660	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
16	6, 15	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
17	16	simpld 495	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
18	17, 10	ffvelcdmd 7087	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
19		onelon 6389	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑋) ∈ 𝐴) → (𝐺‘𝑋) ∈ On)
20	1, 18, 19	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ On)
21		omcl 8535	. . . . 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 7443	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 supp ∅) ∈ V)
24		cantnflem1.o	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))
25	3, 1, 2, 24, 6	cantnfcl 9661	. . . . . . . . . . 11 ⊢ (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
26	25	simpld 495	. . . . . . . . . 10 ⊢ (𝜑 → E We (𝐺 supp ∅))
27	24	oiiso 9531	. . . . . . . . . 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 7319	. . . . . . . . 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 6845	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) → ^◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
32		f1of 6833	. . . . . . . 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 9679	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))
35	33, 34	ffvelcdmd 7087	. . . . . 6 ⊢ (𝜑 → (^◡𝑂‘𝑋) ∈ dom 𝑂)
36	25	simprd 496	. . . . . 6 ⊢ (𝜑 → dom 𝑂 ∈ ω)
37		elnn 7865	. . . . . 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 9659	. . . . . 6 ⊢ 𝐻:ω⟶On
41	40	ffvelcdmi 7085	. . . . 5 ⊢ ((^◡𝑂‘𝑋) ∈ ω → (𝐻‘(^◡𝑂‘𝑋)) ∈ On)
42	38, 41	syl 17	. . . 4 ⊢ (𝜑 → (𝐻‘(^◡𝑂‘𝑋)) ∈ On)
43		oaword1 8551	. . . 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 9664	. . . . 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 7274	. . . . . . . 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 7424	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑_o (𝑂‘(^◡𝑂‘𝑋))) = (𝐴 ↑_o 𝑋))
50	48	fveq2d 6895	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑂‘(^◡𝑂‘𝑋))) = (𝐺‘𝑋))
51	49, 50	oveq12d 7426	. . . . 5 ⊢ (𝜑 → ((𝐴 ↑_o (𝑂‘(^◡𝑂‘𝑋))) ·_o (𝐺‘(𝑂‘(^◡𝑂‘𝑋)))) = ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)))
52	51	oveq1d 7423	. . . 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 2772	. . 3 ⊢ (𝜑 → (𝐻‘suc (^◡𝑂‘𝑋)) = (((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)) +_o (𝐻‘(^◡𝑂‘𝑋))))
54	44, 53	sseqtrrd 4023	. 2 ⊢ (𝜑 → ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)) ⊆ (𝐻‘suc (^◡𝑂‘𝑋)))
55		onss 7771	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
56	2, 55	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ On)
57	56	sselda 3982	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
58	12	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ On)
59		onsseleq 6405	. . . . . . . . 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 868	. . . . . . . 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 4551	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅) = if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅))
64	63	mpteq2dva 5248	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅)) = (𝑥 ∈ 𝐵 ↦ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅)))
65	64	fveq2d 6895	. . . 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 9660	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
67	5, 66	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))
68	67	simpld 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
69	68	ffvelcdmda 7086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ 𝐴)
70	18	ne0d 4335	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≠ ∅)
71		on0eln0 6420	. . . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ 𝐴)
75	69, 74	ifcld 4574	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) ∈ 𝐴)
76	75	fmpttd 7114	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)):𝐵⟶𝐴)
77		0ex 5307	. . . . . . . . 9 ⊢ ∅ ∈ V
78	77	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ V)
79	67	simprd 496	. . . . . . . 8 ⊢ (𝜑 → 𝐹 finSupp ∅)
80	68, 2, 78, 79	fsuppmptif 9393	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) finSupp ∅)
81	3, 1, 2	cantnfs 9660	. . . . . . 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	ffvelcdmd 7087	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐴)
84		eldifn 4127	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∖ 𝑋) → ¬ 𝑦 ∈ 𝑋)
85	84	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ 𝑋)) → ¬ 𝑦 ∈ 𝑋)
86	85	iffalsed 4539	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ 𝑋)) → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) = ∅)
87	86, 2	suppss2 8184	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) supp ∅) ⊆ 𝑋)
88		ifor 4582	. . . . . . . 8 ⊢ if((𝑥 = 𝑋 ∨ 𝑥 ∈ 𝑋), (𝐹‘𝑥), ∅) = if(𝑥 = 𝑋, (𝐹‘𝑥), if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
89		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
90	89	adantl 482	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
91	90	ifeq1da 4559	. . . . . . . . 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 2816	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋))
93		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
94	92, 93	ifbieq1d 4552	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅) = if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
95		eqid 2732	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅)) = (𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))
96		fvex 6904	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑥) ∈ V
97	96, 77	ifex 4578	. . . . . . . . . . 11 ⊢ if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅) ∈ V
98	94, 95, 97	fvmpt 6998	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → ((𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))‘𝑥) = if(𝑥 ∈ 𝑋, (𝐹‘𝑥), ∅))
99	98	ifeq2d 4548	. . . . . . . . 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 2774	. . . . . . . 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 2791	. . . . . . 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 5251	. . . . . 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 9675	. . . . 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 496	. . . 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 2772	. . 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 6389	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐹‘𝑋) ∈ 𝐴) → (𝐹‘𝑋) ∈ On)
107	1, 83, 106	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ On)
108		omsuc 8525	. . . . . 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 6374	. . . . . . . 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 7805	. . . . . . 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 7798	. . . . . . . 8 ⊢ ((𝐹‘𝑋) ∈ On → suc (𝐹‘𝑋) ∈ On)
116	107, 115	syl 17	. . . . . . 7 ⊢ (𝜑 → suc (𝐹‘𝑋) ∈ On)
117		omwordi 8570	. . . . . . 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 1371	. . . . . 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 4021	. . . 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 9667	. . . . 5 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦 ∈ 𝐵 ↦ if(𝑦 ∈ 𝑋, (𝐹‘𝑦), ∅))) ∈ (𝐴 ↑_o 𝑋))
122		onelon 6389	. . . . . . 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 8535	. . . . . . 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 8546	. . . . . 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 1371	. . . . 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 3983	. . 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 2833	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ ((𝐴 ↑_o 𝑋) ·_o (𝐺‘𝑋)))
131	54, 130	sseldd 3983	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 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 Vcvv 3474 ∖ cdif 3945 ⊆ wss 3948 ∅c0 4322 ifcif 4528 ∪ cuni 4908 class class class wbr 5148 {copab 5210 ↦ cmpt 5231 E cep 5579 We wwe 5630 ^◡ccnv 5675 dom cdm 5676 Ord word 6363 Oncon0 6364 suc csuc 6366 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 Isom wiso 6544 (class class class)co 7408 ∈ cmpo 7410 ωcom 7854 supp csupp 8145 seq_ωcseqom 8446 +_o coa 8462 ·_o comu 8463 ↑_o coe 8464 finSupp cfsupp 9360 OrdIsocoi 9503 CNF ccnf 9655

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-seqom 8447 df-1o 8465 df-2o 8466 df-oadd 8469 df-omul 8470 df-oexp 8471 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-oi 9504 df-cnf 9656

This theorem is referenced by: cantnflem1 9683

W3C validator