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Theorem cantnfp1lem3 9601

Description: Lemma for cantnfp1 9602. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)

**Hypotheses**
Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
cantnfp1.g	⊢ (𝜑 → 𝐺 ∈ 𝑆)
cantnfp1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
cantnfp1.y	⊢ (𝜑 → 𝑌 ∈ 𝐴)
cantnfp1.s	⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
cantnfp1.f	⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))
cantnfp1.e	⊢ (𝜑 → ∅ ∈ 𝑌)
cantnfp1.o	⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅))
cantnfp1.h	⊢ 𝐻 = seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_o (𝑂‘𝑘)) ·_o (𝐹‘(𝑂‘𝑘))) +_o 𝑧)), ∅)
cantnfp1.k	⊢ 𝐾 = OrdIso( E , (𝐺 supp ∅))
cantnfp1.m	⊢ 𝑀 = seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_o (𝐾‘𝑘)) ·_o (𝐺‘(𝐾‘𝑘))) +_o 𝑧)), ∅)

**Assertion**
Ref	Expression
cantnfp1lem3	⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o ((𝐴 CNF 𝐵)‘𝐺)))

Distinct variable groups: 𝑡,𝑘,𝑧,𝐵 𝐴,𝑘,𝑡,𝑧 𝑘,𝐹,𝑧 𝑆,𝑘,𝑡,𝑧 𝑘,𝐺,𝑡,𝑧 𝑘,𝐾,𝑡,𝑧 𝑘,𝑂,𝑧 𝜑,𝑘,𝑡,𝑧 𝑘,𝑌,𝑡,𝑧 𝑘,𝑋,𝑡,𝑧 Allowed substitution hints: 𝐹(𝑡) 𝐻(𝑧,𝑡,𝑘) 𝑀(𝑧,𝑡,𝑘) 𝑂(𝑡)

Proof of Theorem cantnfp1lem3 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfs.s	. . 3 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
2		cantnfs.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
3		cantnfs.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
4		cantnfp1.o	. . 3 ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅))
5		cantnfp1.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑆)
6		cantnfp1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		cantnfp1.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
8		cantnfp1.s	. . . 4 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
9		cantnfp1.f	. . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))
10	1, 2, 3, 5, 6, 7, 8, 9	cantnfp1lem1 9599	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
11		cantnfp1.h	. . 3 ⊢ 𝐻 = seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_o (𝑂‘𝑘)) ·_o (𝐹‘(𝑂‘𝑘))) +_o 𝑧)), ∅)
12	1, 2, 3, 4, 10, 11	cantnfval 9589	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝑂))
13		cantnfp1.e	. . . 4 ⊢ (𝜑 → ∅ ∈ 𝑌)
14	1, 2, 3, 5, 6, 7, 8, 9, 13, 4	cantnfp1lem2 9600	. . 3 ⊢ (𝜑 → dom 𝑂 = suc ∪ dom 𝑂)
15	14	fveq2d 6844	. 2 ⊢ (𝜑 → (𝐻‘dom 𝑂) = (𝐻‘suc ∪ dom 𝑂))
16	1, 2, 3, 4, 10	cantnfcl 9588	. . . . . . 7 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝑂 ∈ ω))
17	16	simprd 495	. . . . . 6 ⊢ (𝜑 → dom 𝑂 ∈ ω)
18	14, 17	eqeltrrd 2837	. . . . 5 ⊢ (𝜑 → suc ∪ dom 𝑂 ∈ ω)
19		peano2b 7834	. . . . 5 ⊢ (∪ dom 𝑂 ∈ ω ↔ suc ∪ dom 𝑂 ∈ ω)
20	18, 19	sylibr 234	. . . 4 ⊢ (𝜑 → ∪ dom 𝑂 ∈ ω)
21	1, 2, 3, 4, 10, 11	cantnfsuc 9591	. . . 4 ⊢ ((𝜑 ∧ ∪ dom 𝑂 ∈ ω) → (𝐻‘suc ∪ dom 𝑂) = (((𝐴 ↑_o (𝑂‘∪ dom 𝑂)) ·_o (𝐹‘(𝑂‘∪ dom 𝑂))) +_o (𝐻‘∪ dom 𝑂)))
22	20, 21	mpdan 688	. . 3 ⊢ (𝜑 → (𝐻‘suc ∪ dom 𝑂) = (((𝐴 ↑_o (𝑂‘∪ dom 𝑂)) ·_o (𝐹‘(𝑂‘∪ dom 𝑂))) +_o (𝐻‘∪ dom 𝑂)))
23		ovexd 7402	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V)
24	16	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → E We (𝐹 supp ∅))
25	4	oiiso 9452	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐹 supp ∅)))
26	23, 24, 25	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑂 Isom E , E (dom 𝑂, (𝐹 supp ∅)))
27		isof1o 7278	. . . . . . . . . . . . . 14 ⊢ (𝑂 Isom E , E (dom 𝑂, (𝐹 supp ∅)) → 𝑂:dom 𝑂–1-1-onto→(𝐹 supp ∅))
28	26, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑂:dom 𝑂–1-1-onto→(𝐹 supp ∅))
29		f1ocnv 6792	. . . . . . . . . . . . 13 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐹 supp ∅) → ^◡𝑂:(𝐹 supp ∅)–1-1-onto→dom 𝑂)
30		f1of 6780	. . . . . . . . . . . . 13 ⊢ (^◡𝑂:(𝐹 supp ∅)–1-1-onto→dom 𝑂 → ^◡𝑂:(𝐹 supp ∅)⟶dom 𝑂)
31	28, 29, 30	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ^◡𝑂:(𝐹 supp ∅)⟶dom 𝑂)
32		iftrue 4472	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑋 → if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)) = 𝑌)
33	9, 32, 6, 7	fvmptd3 6971	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝑋) = 𝑌)
34	13	ne0d 4282	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ≠ ∅)
35	33, 34	eqnetrd 2999	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝑋) ≠ ∅)
36	7	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑌 ∈ 𝐴)
37	1, 2, 3	cantnfs 9587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
38	5, 37	mpbid 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
39	38	simpld 494	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
40	39	ffvelcdmda 7036	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐺‘𝑡) ∈ 𝐴)
41	36, 40	ifcld 4513	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)) ∈ 𝐴)
42	41, 9	fmptd 7066	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
43	42	ffnd 6669	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐵)
44		0ex 5242	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
45	44	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∅ ∈ V)
46		elsuppfn 8120	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) → (𝑋 ∈ (𝐹 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ≠ ∅)))
47	43, 3, 45, 46	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ≠ ∅)))
48	6, 35, 47	mpbir2and 714	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (𝐹 supp ∅))
49	31, 48	ffvelcdmd 7037	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝑂‘𝑋) ∈ dom 𝑂)
50		elssuni 4881	. . . . . . . . . . 11 ⊢ ((^◡𝑂‘𝑋) ∈ dom 𝑂 → (^◡𝑂‘𝑋) ⊆ ∪ dom 𝑂)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝑂‘𝑋) ⊆ ∪ dom 𝑂)
52	4	oicl 9444	. . . . . . . . . . . 12 ⊢ Ord dom 𝑂
53		ordelon 6347	. . . . . . . . . . . 12 ⊢ ((Ord dom 𝑂 ∧ (^◡𝑂‘𝑋) ∈ dom 𝑂) → (^◡𝑂‘𝑋) ∈ On)
54	52, 49, 53	sylancr 588	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝑂‘𝑋) ∈ On)
55		nnon 7823	. . . . . . . . . . . 12 ⊢ (∪ dom 𝑂 ∈ ω → ∪ dom 𝑂 ∈ On)
56	20, 55	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ dom 𝑂 ∈ On)
57		ontri1 6357	. . . . . . . . . . 11 ⊢ (((^◡𝑂‘𝑋) ∈ On ∧ ∪ dom 𝑂 ∈ On) → ((^◡𝑂‘𝑋) ⊆ ∪ dom 𝑂 ↔ ¬ ∪ dom 𝑂 ∈ (^◡𝑂‘𝑋)))
58	54, 56, 57	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((^◡𝑂‘𝑋) ⊆ ∪ dom 𝑂 ↔ ¬ ∪ dom 𝑂 ∈ (^◡𝑂‘𝑋)))
59	51, 58	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → ¬ ∪ dom 𝑂 ∈ (^◡𝑂‘𝑋))
60		sucidg 6406	. . . . . . . . . . . . . 14 ⊢ (∪ dom 𝑂 ∈ ω → ∪ dom 𝑂 ∈ suc ∪ dom 𝑂)
61	20, 60	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ dom 𝑂 ∈ suc ∪ dom 𝑂)
62	61, 14	eleqtrrd 2839	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ dom 𝑂 ∈ dom 𝑂)
63		isorel 7281	. . . . . . . . . . . 12 ⊢ ((𝑂 Isom E , E (dom 𝑂, (𝐹 supp ∅)) ∧ (∪ dom 𝑂 ∈ dom 𝑂 ∧ (^◡𝑂‘𝑋) ∈ dom 𝑂)) → (∪ dom 𝑂 E (^◡𝑂‘𝑋) ↔ (𝑂‘∪ dom 𝑂) E (𝑂‘(^◡𝑂‘𝑋))))
64	26, 62, 49, 63	syl12anc 837	. . . . . . . . . . 11 ⊢ (𝜑 → (∪ dom 𝑂 E (^◡𝑂‘𝑋) ↔ (𝑂‘∪ dom 𝑂) E (𝑂‘(^◡𝑂‘𝑋))))
65		fvex 6853	. . . . . . . . . . . 12 ⊢ (^◡𝑂‘𝑋) ∈ V
66	65	epeli 5533	. . . . . . . . . . 11 ⊢ (∪ dom 𝑂 E (^◡𝑂‘𝑋) ↔ ∪ dom 𝑂 ∈ (^◡𝑂‘𝑋))
67		fvex 6853	. . . . . . . . . . . 12 ⊢ (𝑂‘(^◡𝑂‘𝑋)) ∈ V
68	67	epeli 5533	. . . . . . . . . . 11 ⊢ ((𝑂‘∪ dom 𝑂) E (𝑂‘(^◡𝑂‘𝑋)) ↔ (𝑂‘∪ dom 𝑂) ∈ (𝑂‘(^◡𝑂‘𝑋)))
69	64, 66, 68	3bitr3g 313	. . . . . . . . . 10 ⊢ (𝜑 → (∪ dom 𝑂 ∈ (^◡𝑂‘𝑋) ↔ (𝑂‘∪ dom 𝑂) ∈ (𝑂‘(^◡𝑂‘𝑋))))
70		f1ocnvfv2 7232	. . . . . . . . . . . 12 ⊢ ((𝑂:dom 𝑂–1-1-onto→(𝐹 supp ∅) ∧ 𝑋 ∈ (𝐹 supp ∅)) → (𝑂‘(^◡𝑂‘𝑋)) = 𝑋)
71	28, 48, 70	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘(^◡𝑂‘𝑋)) = 𝑋)
72	71	eleq2d 2822	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑂‘∪ dom 𝑂) ∈ (𝑂‘(^◡𝑂‘𝑋)) ↔ (𝑂‘∪ dom 𝑂) ∈ 𝑋))
73	69, 72	bitrd 279	. . . . . . . . 9 ⊢ (𝜑 → (∪ dom 𝑂 ∈ (^◡𝑂‘𝑋) ↔ (𝑂‘∪ dom 𝑂) ∈ 𝑋))
74	59, 73	mtbid 324	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑂‘∪ dom 𝑂) ∈ 𝑋)
75	8	sseld 3920	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑂‘∪ dom 𝑂) ∈ (𝐺 supp ∅) → (𝑂‘∪ dom 𝑂) ∈ 𝑋))
76		suppssdm 8127	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 supp ∅) ⊆ dom 𝐹
77	76, 42	fssdm 6687	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐵)
78		onss 7739	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
79	3, 78	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ⊆ On)
80	77, 79	sstrd 3932	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ On)
81	4	oif 9445	. . . . . . . . . . . . . . . 16 ⊢ 𝑂:dom 𝑂⟶(𝐹 supp ∅)
82	81	ffvelcdmi 7035	. . . . . . . . . . . . . . 15 ⊢ (∪ dom 𝑂 ∈ dom 𝑂 → (𝑂‘∪ dom 𝑂) ∈ (𝐹 supp ∅))
83	62, 82	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘∪ dom 𝑂) ∈ (𝐹 supp ∅))
84	80, 83	sseldd 3922	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘∪ dom 𝑂) ∈ On)
85		eloni 6333	. . . . . . . . . . . . 13 ⊢ ((𝑂‘∪ dom 𝑂) ∈ On → Ord (𝑂‘∪ dom 𝑂))
86	84, 85	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → Ord (𝑂‘∪ dom 𝑂))
87		ordn2lp 6343	. . . . . . . . . . . 12 ⊢ (Ord (𝑂‘∪ dom 𝑂) → ¬ ((𝑂‘∪ dom 𝑂) ∈ 𝑋 ∧ 𝑋 ∈ (𝑂‘∪ dom 𝑂)))
88	86, 87	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ((𝑂‘∪ dom 𝑂) ∈ 𝑋 ∧ 𝑋 ∈ (𝑂‘∪ dom 𝑂)))
89		imnan 399	. . . . . . . . . . 11 ⊢ (((𝑂‘∪ dom 𝑂) ∈ 𝑋 → ¬ 𝑋 ∈ (𝑂‘∪ dom 𝑂)) ↔ ¬ ((𝑂‘∪ dom 𝑂) ∈ 𝑋 ∧ 𝑋 ∈ (𝑂‘∪ dom 𝑂)))
90	88, 89	sylibr 234	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑂‘∪ dom 𝑂) ∈ 𝑋 → ¬ 𝑋 ∈ (𝑂‘∪ dom 𝑂)))
91	75, 90	syld 47	. . . . . . . . 9 ⊢ (𝜑 → ((𝑂‘∪ dom 𝑂) ∈ (𝐺 supp ∅) → ¬ 𝑋 ∈ (𝑂‘∪ dom 𝑂)))
92		onelon 6348	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
93	3, 6, 92	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ On)
94		eloni 6333	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ On → Ord 𝑋)
95	93, 94	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Ord 𝑋)
96		ordirr 6341	. . . . . . . . . . 11 ⊢ (Ord 𝑋 → ¬ 𝑋 ∈ 𝑋)
97	95, 96	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑋)
98		elsni 4584	. . . . . . . . . . . 12 ⊢ ((𝑂‘∪ dom 𝑂) ∈ {𝑋} → (𝑂‘∪ dom 𝑂) = 𝑋)
99	98	eleq2d 2822	. . . . . . . . . . 11 ⊢ ((𝑂‘∪ dom 𝑂) ∈ {𝑋} → (𝑋 ∈ (𝑂‘∪ dom 𝑂) ↔ 𝑋 ∈ 𝑋))
100	99	notbid 318	. . . . . . . . . 10 ⊢ ((𝑂‘∪ dom 𝑂) ∈ {𝑋} → (¬ 𝑋 ∈ (𝑂‘∪ dom 𝑂) ↔ ¬ 𝑋 ∈ 𝑋))
101	97, 100	syl5ibrcom 247	. . . . . . . . 9 ⊢ (𝜑 → ((𝑂‘∪ dom 𝑂) ∈ {𝑋} → ¬ 𝑋 ∈ (𝑂‘∪ dom 𝑂)))
102		eqeq1 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑘 → (𝑡 = 𝑋 ↔ 𝑘 = 𝑋))
103		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑘 → (𝐺‘𝑡) = (𝐺‘𝑘))
104	102, 103	ifbieq2d 4493	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑘 → if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)) = if(𝑘 = 𝑋, 𝑌, (𝐺‘𝑘)))
105		eldifi 4071	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋})) → 𝑘 ∈ 𝐵)
106	105	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋}))) → 𝑘 ∈ 𝐵)
107	7	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋}))) → 𝑌 ∈ 𝐴)
108		fvex 6853	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘𝑘) ∈ V
109		ifexg 4516	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ∈ 𝐴 ∧ (𝐺‘𝑘) ∈ V) → if(𝑘 = 𝑋, 𝑌, (𝐺‘𝑘)) ∈ V)
110	107, 108, 109	sylancl 587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋}))) → if(𝑘 = 𝑋, 𝑌, (𝐺‘𝑘)) ∈ V)
111	9, 104, 106, 110	fvmptd3 6971	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋}))) → (𝐹‘𝑘) = if(𝑘 = 𝑋, 𝑌, (𝐺‘𝑘)))
112		eldifn 4072	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋})) → ¬ 𝑘 ∈ ((𝐺 supp ∅) ∪ {𝑋}))
113	112	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋}))) → ¬ 𝑘 ∈ ((𝐺 supp ∅) ∪ {𝑋}))
114		velsn 4583	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ {𝑋} ↔ 𝑘 = 𝑋)
115		elun2 4123	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ {𝑋} → 𝑘 ∈ ((𝐺 supp ∅) ∪ {𝑋}))
116	114, 115	sylbir 235	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑋 → 𝑘 ∈ ((𝐺 supp ∅) ∪ {𝑋}))
117	113, 116	nsyl 140	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋}))) → ¬ 𝑘 = 𝑋)
118	117	iffalsed 4477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋}))) → if(𝑘 = 𝑋, 𝑌, (𝐺‘𝑘)) = (𝐺‘𝑘))
119		ssun1 4118	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 supp ∅) ⊆ ((𝐺 supp ∅) ∪ {𝑋})
120		sscon 4083	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 supp ∅) ⊆ ((𝐺 supp ∅) ∪ {𝑋}) → (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋})) ⊆ (𝐵 ∖ (𝐺 supp ∅)))
121	119, 120	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋})) ⊆ (𝐵 ∖ (𝐺 supp ∅))
122	121	sseli 3917	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋})) → 𝑘 ∈ (𝐵 ∖ (𝐺 supp ∅)))
123		ssidd 3945	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ (𝐺 supp ∅))
124	39, 123, 3, 13	suppssr 8145	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐺 supp ∅))) → (𝐺‘𝑘) = ∅)
125	122, 124	sylan2 594	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋}))) → (𝐺‘𝑘) = ∅)
126	111, 118, 125	3eqtrd 2775	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ ((𝐺 supp ∅) ∪ {𝑋}))) → (𝐹‘𝑘) = ∅)
127	42, 126	suppss 8144	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ ((𝐺 supp ∅) ∪ {𝑋}))
128	127, 83	sseldd 3922	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘∪ dom 𝑂) ∈ ((𝐺 supp ∅) ∪ {𝑋}))
129		elun 4093	. . . . . . . . . 10 ⊢ ((𝑂‘∪ dom 𝑂) ∈ ((𝐺 supp ∅) ∪ {𝑋}) ↔ ((𝑂‘∪ dom 𝑂) ∈ (𝐺 supp ∅) ∨ (𝑂‘∪ dom 𝑂) ∈ {𝑋}))
130	128, 129	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → ((𝑂‘∪ dom 𝑂) ∈ (𝐺 supp ∅) ∨ (𝑂‘∪ dom 𝑂) ∈ {𝑋}))
131	91, 101, 130	mpjaod 861	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑂‘∪ dom 𝑂))
132		ioran 986	. . . . . . . 8 ⊢ (¬ ((𝑂‘∪ dom 𝑂) ∈ 𝑋 ∨ 𝑋 ∈ (𝑂‘∪ dom 𝑂)) ↔ (¬ (𝑂‘∪ dom 𝑂) ∈ 𝑋 ∧ ¬ 𝑋 ∈ (𝑂‘∪ dom 𝑂)))
133	74, 131, 132	sylanbrc 584	. . . . . . 7 ⊢ (𝜑 → ¬ ((𝑂‘∪ dom 𝑂) ∈ 𝑋 ∨ 𝑋 ∈ (𝑂‘∪ dom 𝑂)))
134		ordtri3 6359	. . . . . . . 8 ⊢ ((Ord (𝑂‘∪ dom 𝑂) ∧ Ord 𝑋) → ((𝑂‘∪ dom 𝑂) = 𝑋 ↔ ¬ ((𝑂‘∪ dom 𝑂) ∈ 𝑋 ∨ 𝑋 ∈ (𝑂‘∪ dom 𝑂))))
135	86, 95, 134	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ((𝑂‘∪ dom 𝑂) = 𝑋 ↔ ¬ ((𝑂‘∪ dom 𝑂) ∈ 𝑋 ∨ 𝑋 ∈ (𝑂‘∪ dom 𝑂))))
136	133, 135	mpbird 257	. . . . . 6 ⊢ (𝜑 → (𝑂‘∪ dom 𝑂) = 𝑋)
137	136	oveq2d 7383	. . . . 5 ⊢ (𝜑 → (𝐴 ↑_o (𝑂‘∪ dom 𝑂)) = (𝐴 ↑_o 𝑋))
138	136	fveq2d 6844	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝑂‘∪ dom 𝑂)) = (𝐹‘𝑋))
139	138, 33	eqtrd 2771	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝑂‘∪ dom 𝑂)) = 𝑌)
140	137, 139	oveq12d 7385	. . . 4 ⊢ (𝜑 → ((𝐴 ↑_o (𝑂‘∪ dom 𝑂)) ·_o (𝐹‘(𝑂‘∪ dom 𝑂))) = ((𝐴 ↑_o 𝑋) ·_o 𝑌))
141		nnord 7825	. . . . . . . . 9 ⊢ (∪ dom 𝑂 ∈ ω → Ord ∪ dom 𝑂)
142	20, 141	syl 17	. . . . . . . 8 ⊢ (𝜑 → Ord ∪ dom 𝑂)
143		sssucid 6405	. . . . . . . . . 10 ⊢ ∪ dom 𝑂 ⊆ suc ∪ dom 𝑂
144	143, 14	sseqtrrid 3965	. . . . . . . . 9 ⊢ (𝜑 → ∪ dom 𝑂 ⊆ dom 𝑂)
145		f1ofo 6787	. . . . . . . . . . . . 13 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐹 supp ∅) → 𝑂:dom 𝑂–onto→(𝐹 supp ∅))
146	28, 145	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂:dom 𝑂–onto→(𝐹 supp ∅))
147		foima 6757	. . . . . . . . . . . 12 ⊢ (𝑂:dom 𝑂–onto→(𝐹 supp ∅) → (𝑂 “ dom 𝑂) = (𝐹 supp ∅))
148	146, 147	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂 “ dom 𝑂) = (𝐹 supp ∅))
149		ffn 6668	. . . . . . . . . . . . . 14 ⊢ (𝑂:dom 𝑂⟶(𝐹 supp ∅) → 𝑂 Fn dom 𝑂)
150	81, 149	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 𝑂 Fn dom 𝑂
151		fnsnfv 6919	. . . . . . . . . . . . 13 ⊢ ((𝑂 Fn dom 𝑂 ∧ ∪ dom 𝑂 ∈ dom 𝑂) → {(𝑂‘∪ dom 𝑂)} = (𝑂 “ {∪ dom 𝑂}))
152	150, 62, 151	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝜑 → {(𝑂‘∪ dom 𝑂)} = (𝑂 “ {∪ dom 𝑂}))
153	136	sneqd 4579	. . . . . . . . . . . 12 ⊢ (𝜑 → {(𝑂‘∪ dom 𝑂)} = {𝑋})
154	152, 153	eqtr3d 2773	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂 “ {∪ dom 𝑂}) = {𝑋})
155	148, 154	difeq12d 4067	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑂 “ dom 𝑂) ∖ (𝑂 “ {∪ dom 𝑂})) = ((𝐹 supp ∅) ∖ {𝑋}))
156		ordirr 6341	. . . . . . . . . . . . . . . . 17 ⊢ (Ord ∪ dom 𝑂 → ¬ ∪ dom 𝑂 ∈ ∪ dom 𝑂)
157	142, 156	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ ∪ dom 𝑂 ∈ ∪ dom 𝑂)
158		disjsn 4655	. . . . . . . . . . . . . . . 16 ⊢ ((∪ dom 𝑂 ∩ {∪ dom 𝑂}) = ∅ ↔ ¬ ∪ dom 𝑂 ∈ ∪ dom 𝑂)
159	157, 158	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∪ dom 𝑂 ∩ {∪ dom 𝑂}) = ∅)
160		disj3 4394	. . . . . . . . . . . . . . 15 ⊢ ((∪ dom 𝑂 ∩ {∪ dom 𝑂}) = ∅ ↔ ∪ dom 𝑂 = (∪ dom 𝑂 ∖ {∪ dom 𝑂}))
161	159, 160	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∪ dom 𝑂 = (∪ dom 𝑂 ∖ {∪ dom 𝑂}))
162		difun2 4421	. . . . . . . . . . . . . 14 ⊢ ((∪ dom 𝑂 ∪ {∪ dom 𝑂}) ∖ {∪ dom 𝑂}) = (∪ dom 𝑂 ∖ {∪ dom 𝑂})
163	161, 162	eqtr4di 2789	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ dom 𝑂 = ((∪ dom 𝑂 ∪ {∪ dom 𝑂}) ∖ {∪ dom 𝑂}))
164		df-suc 6329	. . . . . . . . . . . . . . 15 ⊢ suc ∪ dom 𝑂 = (∪ dom 𝑂 ∪ {∪ dom 𝑂})
165	14, 164	eqtrdi 2787	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑂 = (∪ dom 𝑂 ∪ {∪ dom 𝑂}))
166	165	difeq1d 4065	. . . . . . . . . . . . 13 ⊢ (𝜑 → (dom 𝑂 ∖ {∪ dom 𝑂}) = ((∪ dom 𝑂 ∪ {∪ dom 𝑂}) ∖ {∪ dom 𝑂}))
167	163, 166	eqtr4d 2774	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ dom 𝑂 = (dom 𝑂 ∖ {∪ dom 𝑂}))
168	167	imaeq2d 6025	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂 “ ∪ dom 𝑂) = (𝑂 “ (dom 𝑂 ∖ {∪ dom 𝑂})))
169		dff1o3 6786	. . . . . . . . . . . . 13 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐹 supp ∅) ↔ (𝑂:dom 𝑂–onto→(𝐹 supp ∅) ∧ Fun ^◡𝑂))
170	169	simprbi 497	. . . . . . . . . . . 12 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐹 supp ∅) → Fun ^◡𝑂)
171		imadif 6582	. . . . . . . . . . . 12 ⊢ (Fun ^◡𝑂 → (𝑂 “ (dom 𝑂 ∖ {∪ dom 𝑂})) = ((𝑂 “ dom 𝑂) ∖ (𝑂 “ {∪ dom 𝑂})))
172	28, 170, 171	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂 “ (dom 𝑂 ∖ {∪ dom 𝑂})) = ((𝑂 “ dom 𝑂) ∖ (𝑂 “ {∪ dom 𝑂})))
173	168, 172	eqtrd 2771	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂 “ ∪ dom 𝑂) = ((𝑂 “ dom 𝑂) ∖ (𝑂 “ {∪ dom 𝑂})))
174	8, 97	ssneldd 3924	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝐺 supp ∅))
175		disjsn 4655	. . . . . . . . . . . . 13 ⊢ (((𝐺 supp ∅) ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ (𝐺 supp ∅))
176	174, 175	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐺 supp ∅) ∩ {𝑋}) = ∅)
177		fvex 6853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺‘𝑋) ∈ V
178		dif1o 8435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺‘𝑋) ∈ (V ∖ 1_o) ↔ ((𝐺‘𝑋) ∈ V ∧ (𝐺‘𝑋) ≠ ∅))
179	177, 178	mpbiran 710	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺‘𝑋) ∈ (V ∖ 1_o) ↔ (𝐺‘𝑋) ≠ ∅)
180	39	ffnd 6669	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐺 Fn 𝐵)
181		elsuppfn 8120	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅)))
182	180, 3, 45, 181	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅)))
183	179	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((𝐺‘𝑋) ∈ (V ∖ 1_o) ↔ (𝐺‘𝑋) ≠ ∅))
184	183	bicomd 223	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝐺‘𝑋) ≠ ∅ ↔ (𝐺‘𝑋) ∈ (V ∖ 1_o)))
185	184	anbi2d 631	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ∈ (V ∖ 1_o))))
186	182, 185	bitrd 279	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ∈ (V ∖ 1_o))))
187	8	sseld 3920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑋 ∈ (𝐺 supp ∅) → 𝑋 ∈ 𝑋))
188	186, 187	sylbird 260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ∈ (V ∖ 1_o)) → 𝑋 ∈ 𝑋))
189	6, 188	mpand 696	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐺‘𝑋) ∈ (V ∖ 1_o) → 𝑋 ∈ 𝑋))
190	179, 189	biimtrrid 243	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐺‘𝑋) ≠ ∅ → 𝑋 ∈ 𝑋))
191	190	necon1bd 2950	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (¬ 𝑋 ∈ 𝑋 → (𝐺‘𝑋) = ∅))
192	97, 191	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐺‘𝑋) = ∅)
193	192	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐹 supp ∅))) → (𝐺‘𝑋) = ∅)
194		fveqeq2 6849	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑋 → ((𝐺‘𝑘) = ∅ ↔ (𝐺‘𝑋) = ∅))
195	193, 194	syl5ibrcom 247	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐹 supp ∅))) → (𝑘 = 𝑋 → (𝐺‘𝑘) = ∅))
196		eldifi 4071	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝐵 ∖ (𝐹 supp ∅)) → 𝑘 ∈ 𝐵)
197	196	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐹 supp ∅))) → 𝑘 ∈ 𝐵)
198	7	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐹 supp ∅))) → 𝑌 ∈ 𝐴)
199	198, 108, 109	sylancl 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐹 supp ∅))) → if(𝑘 = 𝑋, 𝑌, (𝐺‘𝑘)) ∈ V)
200	9, 104, 197, 199	fvmptd3 6971	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐹 supp ∅))) → (𝐹‘𝑘) = if(𝑘 = 𝑋, 𝑌, (𝐺‘𝑘)))
201		ssidd 3945	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ (𝐹 supp ∅))
202	42, 201, 3, 13	suppssr 8145	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐹 supp ∅))) → (𝐹‘𝑘) = ∅)
203	200, 202	eqtr3d 2773	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐹 supp ∅))) → if(𝑘 = 𝑋, 𝑌, (𝐺‘𝑘)) = ∅)
204		iffalse 4475	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑌, (𝐺‘𝑘)) = (𝐺‘𝑘))
205	204	eqeq1d 2738	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑌, (𝐺‘𝑘)) = ∅ ↔ (𝐺‘𝑘) = ∅))
206	203, 205	syl5ibcom 245	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐹 supp ∅))) → (¬ 𝑘 = 𝑋 → (𝐺‘𝑘) = ∅))
207	195, 206	pm2.61d 179	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐹 supp ∅))) → (𝐺‘𝑘) = ∅)
208	39, 207	suppss 8144	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ (𝐹 supp ∅))
209		reldisj 4393	. . . . . . . . . . . . 13 ⊢ ((𝐺 supp ∅) ⊆ (𝐹 supp ∅) → (((𝐺 supp ∅) ∩ {𝑋}) = ∅ ↔ (𝐺 supp ∅) ⊆ ((𝐹 supp ∅) ∖ {𝑋})))
210	208, 209	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐺 supp ∅) ∩ {𝑋}) = ∅ ↔ (𝐺 supp ∅) ⊆ ((𝐹 supp ∅) ∖ {𝑋})))
211	176, 210	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ ((𝐹 supp ∅) ∖ {𝑋}))
212		uncom 4098	. . . . . . . . . . . . 13 ⊢ ((𝐺 supp ∅) ∪ {𝑋}) = ({𝑋} ∪ (𝐺 supp ∅))
213	127, 212	sseqtrdi 3962	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ ({𝑋} ∪ (𝐺 supp ∅)))
214		ssundif 4427	. . . . . . . . . . . 12 ⊢ ((𝐹 supp ∅) ⊆ ({𝑋} ∪ (𝐺 supp ∅)) ↔ ((𝐹 supp ∅) ∖ {𝑋}) ⊆ (𝐺 supp ∅))
215	213, 214	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 supp ∅) ∖ {𝑋}) ⊆ (𝐺 supp ∅))
216	211, 215	eqssd 3939	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 supp ∅) = ((𝐹 supp ∅) ∖ {𝑋}))
217	155, 173, 216	3eqtr4rd 2782	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 supp ∅) = (𝑂 “ ∪ dom 𝑂))
218		isores3 7290	. . . . . . . . 9 ⊢ ((𝑂 Isom E , E (dom 𝑂, (𝐹 supp ∅)) ∧ ∪ dom 𝑂 ⊆ dom 𝑂 ∧ (𝐺 supp ∅) = (𝑂 “ ∪ dom 𝑂)) → (𝑂 ↾ ∪ dom 𝑂) Isom E , E (∪ dom 𝑂, (𝐺 supp ∅)))
219	26, 144, 217, 218	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (𝑂 ↾ ∪ dom 𝑂) Isom E , E (∪ dom 𝑂, (𝐺 supp ∅)))
220		cantnfp1.k	. . . . . . . . . . 11 ⊢ 𝐾 = OrdIso( E , (𝐺 supp ∅))
221	1, 2, 3, 220, 5	cantnfcl 9588	. . . . . . . . . 10 ⊢ (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝐾 ∈ ω))
222	221	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → E We (𝐺 supp ∅))
223		epse 5613	. . . . . . . . 9 ⊢ E Se (𝐺 supp ∅)
224	220	oieu 9454	. . . . . . . . 9 ⊢ (( E We (𝐺 supp ∅) ∧ E Se (𝐺 supp ∅)) → ((Ord ∪ dom 𝑂 ∧ (𝑂 ↾ ∪ dom 𝑂) Isom E , E (∪ dom 𝑂, (𝐺 supp ∅))) ↔ (∪ dom 𝑂 = dom 𝐾 ∧ (𝑂 ↾ ∪ dom 𝑂) = 𝐾)))
225	222, 223, 224	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → ((Ord ∪ dom 𝑂 ∧ (𝑂 ↾ ∪ dom 𝑂) Isom E , E (∪ dom 𝑂, (𝐺 supp ∅))) ↔ (∪ dom 𝑂 = dom 𝐾 ∧ (𝑂 ↾ ∪ dom 𝑂) = 𝐾)))
226	142, 219, 225	mpbi2and 713	. . . . . . 7 ⊢ (𝜑 → (∪ dom 𝑂 = dom 𝐾 ∧ (𝑂 ↾ ∪ dom 𝑂) = 𝐾))
227	226	simpld 494	. . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 = dom 𝐾)
228	227	fveq2d 6844	. . . . 5 ⊢ (𝜑 → (𝑀‘∪ dom 𝑂) = (𝑀‘dom 𝐾))
229		eleq1 2824	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑥 ∈ dom 𝑂 ↔ ∅ ∈ dom 𝑂))
230		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝐻‘𝑥) = (𝐻‘∅))
231		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑀‘𝑥) = (𝑀‘∅))
232		cantnfp1.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_o (𝐾‘𝑘)) ·_o (𝐺‘(𝐾‘𝑘))) +_o 𝑧)), ∅)
233	232	seqom0g 8395	. . . . . . . . . . . . 13 ⊢ (∅ ∈ V → (𝑀‘∅) = ∅)
234	44, 233	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑀‘∅) = ∅
235	231, 234	eqtrdi 2787	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑀‘𝑥) = ∅)
236	230, 235	eqeq12d 2752	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝐻‘𝑥) = (𝑀‘𝑥) ↔ (𝐻‘∅) = ∅))
237	229, 236	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝑥 ∈ dom 𝑂 → (𝐻‘𝑥) = (𝑀‘𝑥)) ↔ (∅ ∈ dom 𝑂 → (𝐻‘∅) = ∅)))
238	237	imbi2d 340	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝜑 → (𝑥 ∈ dom 𝑂 → (𝐻‘𝑥) = (𝑀‘𝑥))) ↔ (𝜑 → (∅ ∈ dom 𝑂 → (𝐻‘∅) = ∅))))
239		eleq1 2824	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑂 ↔ 𝑦 ∈ dom 𝑂))
240		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐻‘𝑥) = (𝐻‘𝑦))
241		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑀‘𝑥) = (𝑀‘𝑦))
242	240, 241	eqeq12d 2752	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐻‘𝑥) = (𝑀‘𝑥) ↔ (𝐻‘𝑦) = (𝑀‘𝑦)))
243	239, 242	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑂 → (𝐻‘𝑥) = (𝑀‘𝑥)) ↔ (𝑦 ∈ dom 𝑂 → (𝐻‘𝑦) = (𝑀‘𝑦))))
244	243	imbi2d 340	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝑥 ∈ dom 𝑂 → (𝐻‘𝑥) = (𝑀‘𝑥))) ↔ (𝜑 → (𝑦 ∈ dom 𝑂 → (𝐻‘𝑦) = (𝑀‘𝑦)))))
245		eleq1 2824	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ dom 𝑂 ↔ suc 𝑦 ∈ dom 𝑂))
246		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (𝐻‘𝑥) = (𝐻‘suc 𝑦))
247		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (𝑀‘𝑥) = (𝑀‘suc 𝑦))
248	246, 247	eqeq12d 2752	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → ((𝐻‘𝑥) = (𝑀‘𝑥) ↔ (𝐻‘suc 𝑦) = (𝑀‘suc 𝑦)))
249	245, 248	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ∈ dom 𝑂 → (𝐻‘𝑥) = (𝑀‘𝑥)) ↔ (suc 𝑦 ∈ dom 𝑂 → (𝐻‘suc 𝑦) = (𝑀‘suc 𝑦))))
250	249	imbi2d 340	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((𝜑 → (𝑥 ∈ dom 𝑂 → (𝐻‘𝑥) = (𝑀‘𝑥))) ↔ (𝜑 → (suc 𝑦 ∈ dom 𝑂 → (𝐻‘suc 𝑦) = (𝑀‘suc 𝑦)))))
251		eleq1 2824	. . . . . . . . . 10 ⊢ (𝑥 = ∪ dom 𝑂 → (𝑥 ∈ dom 𝑂 ↔ ∪ dom 𝑂 ∈ dom 𝑂))
252		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑥 = ∪ dom 𝑂 → (𝐻‘𝑥) = (𝐻‘∪ dom 𝑂))
253		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑥 = ∪ dom 𝑂 → (𝑀‘𝑥) = (𝑀‘∪ dom 𝑂))
254	252, 253	eqeq12d 2752	. . . . . . . . . 10 ⊢ (𝑥 = ∪ dom 𝑂 → ((𝐻‘𝑥) = (𝑀‘𝑥) ↔ (𝐻‘∪ dom 𝑂) = (𝑀‘∪ dom 𝑂)))
255	251, 254	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = ∪ dom 𝑂 → ((𝑥 ∈ dom 𝑂 → (𝐻‘𝑥) = (𝑀‘𝑥)) ↔ (∪ dom 𝑂 ∈ dom 𝑂 → (𝐻‘∪ dom 𝑂) = (𝑀‘∪ dom 𝑂))))
256	255	imbi2d 340	. . . . . . . 8 ⊢ (𝑥 = ∪ dom 𝑂 → ((𝜑 → (𝑥 ∈ dom 𝑂 → (𝐻‘𝑥) = (𝑀‘𝑥))) ↔ (𝜑 → (∪ dom 𝑂 ∈ dom 𝑂 → (𝐻‘∪ dom 𝑂) = (𝑀‘∪ dom 𝑂)))))
257	11	seqom0g 8395	. . . . . . . . 9 ⊢ (∅ ∈ dom 𝑂 → (𝐻‘∅) = ∅)
258	257	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (∅ ∈ dom 𝑂 → (𝐻‘∅) = ∅))
259		nnord 7825	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑂 ∈ ω → Ord dom 𝑂)
260	17, 259	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Ord dom 𝑂)
261		ordtr 6337	. . . . . . . . . . . . . . 15 ⊢ (Ord dom 𝑂 → Tr dom 𝑂)
262	260, 261	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Tr dom 𝑂)
263		trsuc 6412	. . . . . . . . . . . . . 14 ⊢ ((Tr dom 𝑂 ∧ suc 𝑦 ∈ dom 𝑂) → 𝑦 ∈ dom 𝑂)
264	262, 263	sylan 581	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → 𝑦 ∈ dom 𝑂)
265	264	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → (suc 𝑦 ∈ dom 𝑂 → 𝑦 ∈ dom 𝑂))
266	265	imim1d 82	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 ∈ dom 𝑂 → (𝐻‘𝑦) = (𝑀‘𝑦)) → (suc 𝑦 ∈ dom 𝑂 → (𝐻‘𝑦) = (𝑀‘𝑦))))
267		oveq2 7375	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑦) = (𝑀‘𝑦) → (((𝐴 ↑_o (𝑂‘𝑦)) ·_o (𝐹‘(𝑂‘𝑦))) +_o (𝐻‘𝑦)) = (((𝐴 ↑_o (𝑂‘𝑦)) ·_o (𝐹‘(𝑂‘𝑦))) +_o (𝑀‘𝑦)))
268		elnn 7828	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → 𝑦 ∈ ω)
269	268	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((dom 𝑂 ∈ ω ∧ 𝑦 ∈ dom 𝑂) → 𝑦 ∈ ω)
270	17, 264, 269	syl2an2r 686	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → 𝑦 ∈ ω)
271	1, 2, 3, 4, 10, 11	cantnfsuc 9591	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (𝐻‘suc 𝑦) = (((𝐴 ↑_o (𝑂‘𝑦)) ·_o (𝐹‘(𝑂‘𝑦))) +_o (𝐻‘𝑦)))
272	270, 271	syldan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (𝐻‘suc 𝑦) = (((𝐴 ↑_o (𝑂‘𝑦)) ·_o (𝐹‘(𝑂‘𝑦))) +_o (𝐻‘𝑦)))
273	1, 2, 3, 220, 5, 232	cantnfsuc 9591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (𝑀‘suc 𝑦) = (((𝐴 ↑_o (𝐾‘𝑦)) ·_o (𝐺‘(𝐾‘𝑦))) +_o (𝑀‘𝑦)))
274	270, 273	syldan 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (𝑀‘suc 𝑦) = (((𝐴 ↑_o (𝐾‘𝑦)) ·_o (𝐺‘(𝐾‘𝑦))) +_o (𝑀‘𝑦)))
275	226	simprd 495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑂 ↾ ∪ dom 𝑂) = 𝐾)
276	275	fveq1d 6842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑂 ↾ ∪ dom 𝑂)‘𝑦) = (𝐾‘𝑦))
277	276	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → ((𝑂 ↾ ∪ dom 𝑂)‘𝑦) = (𝐾‘𝑦))
278	14	eleq2d 2822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (suc 𝑦 ∈ dom 𝑂 ↔ suc 𝑦 ∈ suc ∪ dom 𝑂))
279	278	biimpa 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → suc 𝑦 ∈ suc ∪ dom 𝑂)
280	142	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → Ord ∪ dom 𝑂)
281		ordsucelsuc 7773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Ord ∪ dom 𝑂 → (𝑦 ∈ ∪ dom 𝑂 ↔ suc 𝑦 ∈ suc ∪ dom 𝑂))
282	280, 281	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (𝑦 ∈ ∪ dom 𝑂 ↔ suc 𝑦 ∈ suc ∪ dom 𝑂))
283	279, 282	mpbird 257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → 𝑦 ∈ ∪ dom 𝑂)
284	283	fvresd 6860	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → ((𝑂 ↾ ∪ dom 𝑂)‘𝑦) = (𝑂‘𝑦))
285	277, 284	eqtr3d 2773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (𝐾‘𝑦) = (𝑂‘𝑦))
286	285	oveq2d 7383	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (𝐴 ↑_o (𝐾‘𝑦)) = (𝐴 ↑_o (𝑂‘𝑦)))
287		eqeq1 2740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = (𝐾‘𝑦) → (𝑡 = 𝑋 ↔ (𝐾‘𝑦) = 𝑋))
288		fveq2 6840	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = (𝐾‘𝑦) → (𝐺‘𝑡) = (𝐺‘(𝐾‘𝑦)))
289	287, 288	ifbieq2d 4493	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝐾‘𝑦) → if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)) = if((𝐾‘𝑦) = 𝑋, 𝑌, (𝐺‘(𝐾‘𝑦))))
290		suppssdm 8127	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐺 supp ∅) ⊆ dom 𝐺
291	290, 39	fssdm 6687	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
292	291	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (𝐺 supp ∅) ⊆ 𝐵)
293	227	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → ∪ dom 𝑂 = dom 𝐾)
294	283, 293	eleqtrd 2838	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → 𝑦 ∈ dom 𝐾)
295	220	oif 9445	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐾:dom 𝐾⟶(𝐺 supp ∅)
296	295	ffvelcdmi 7035	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ dom 𝐾 → (𝐾‘𝑦) ∈ (𝐺 supp ∅))
297	294, 296	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (𝐾‘𝑦) ∈ (𝐺 supp ∅))
298	292, 297	sseldd 3922	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (𝐾‘𝑦) ∈ 𝐵)
299	7	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → 𝑌 ∈ 𝐴)
300		fvex 6853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺‘(𝐾‘𝑦)) ∈ V
301		ifexg 4516	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑌 ∈ 𝐴 ∧ (𝐺‘(𝐾‘𝑦)) ∈ V) → if((𝐾‘𝑦) = 𝑋, 𝑌, (𝐺‘(𝐾‘𝑦))) ∈ V)
302	299, 300, 301	sylancl 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → if((𝐾‘𝑦) = 𝑋, 𝑌, (𝐺‘(𝐾‘𝑦))) ∈ V)
303	9, 289, 298, 302	fvmptd3 6971	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (𝐹‘(𝐾‘𝑦)) = if((𝐾‘𝑦) = 𝑋, 𝑌, (𝐺‘(𝐾‘𝑦))))
304	285	fveq2d 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (𝐹‘(𝐾‘𝑦)) = (𝐹‘(𝑂‘𝑦)))
305	174	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → ¬ 𝑋 ∈ (𝐺 supp ∅))
306		nelneq 2860	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐾‘𝑦) ∈ (𝐺 supp ∅) ∧ ¬ 𝑋 ∈ (𝐺 supp ∅)) → ¬ (𝐾‘𝑦) = 𝑋)
307	297, 305, 306	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → ¬ (𝐾‘𝑦) = 𝑋)
308	307	iffalsed 4477	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → if((𝐾‘𝑦) = 𝑋, 𝑌, (𝐺‘(𝐾‘𝑦))) = (𝐺‘(𝐾‘𝑦)))
309	303, 304, 308	3eqtr3rd 2780	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (𝐺‘(𝐾‘𝑦)) = (𝐹‘(𝑂‘𝑦)))
310	286, 309	oveq12d 7385	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → ((𝐴 ↑_o (𝐾‘𝑦)) ·_o (𝐺‘(𝐾‘𝑦))) = ((𝐴 ↑_o (𝑂‘𝑦)) ·_o (𝐹‘(𝑂‘𝑦))))
311	310	oveq1d 7382	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (((𝐴 ↑_o (𝐾‘𝑦)) ·_o (𝐺‘(𝐾‘𝑦))) +_o (𝑀‘𝑦)) = (((𝐴 ↑_o (𝑂‘𝑦)) ·_o (𝐹‘(𝑂‘𝑦))) +_o (𝑀‘𝑦)))
312	274, 311	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → (𝑀‘suc 𝑦) = (((𝐴 ↑_o (𝑂‘𝑦)) ·_o (𝐹‘(𝑂‘𝑦))) +_o (𝑀‘𝑦)))
313	272, 312	eqeq12d 2752	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → ((𝐻‘suc 𝑦) = (𝑀‘suc 𝑦) ↔ (((𝐴 ↑_o (𝑂‘𝑦)) ·_o (𝐹‘(𝑂‘𝑦))) +_o (𝐻‘𝑦)) = (((𝐴 ↑_o (𝑂‘𝑦)) ·_o (𝐹‘(𝑂‘𝑦))) +_o (𝑀‘𝑦))))
314	267, 313	imbitrrid 246	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ suc 𝑦 ∈ dom 𝑂) → ((𝐻‘𝑦) = (𝑀‘𝑦) → (𝐻‘suc 𝑦) = (𝑀‘suc 𝑦)))
315	314	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → (suc 𝑦 ∈ dom 𝑂 → ((𝐻‘𝑦) = (𝑀‘𝑦) → (𝐻‘suc 𝑦) = (𝑀‘suc 𝑦))))
316	315	a2d 29	. . . . . . . . . . 11 ⊢ (𝜑 → ((suc 𝑦 ∈ dom 𝑂 → (𝐻‘𝑦) = (𝑀‘𝑦)) → (suc 𝑦 ∈ dom 𝑂 → (𝐻‘suc 𝑦) = (𝑀‘suc 𝑦))))
317	266, 316	syld 47	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ dom 𝑂 → (𝐻‘𝑦) = (𝑀‘𝑦)) → (suc 𝑦 ∈ dom 𝑂 → (𝐻‘suc 𝑦) = (𝑀‘suc 𝑦))))
318	317	a2i 14	. . . . . . . . 9 ⊢ ((𝜑 → (𝑦 ∈ dom 𝑂 → (𝐻‘𝑦) = (𝑀‘𝑦))) → (𝜑 → (suc 𝑦 ∈ dom 𝑂 → (𝐻‘suc 𝑦) = (𝑀‘suc 𝑦))))
319	318	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝜑 → (𝑦 ∈ dom 𝑂 → (𝐻‘𝑦) = (𝑀‘𝑦))) → (𝜑 → (suc 𝑦 ∈ dom 𝑂 → (𝐻‘suc 𝑦) = (𝑀‘suc 𝑦)))))
320	238, 244, 250, 256, 258, 319	finds 7847	. . . . . . 7 ⊢ (∪ dom 𝑂 ∈ ω → (𝜑 → (∪ dom 𝑂 ∈ dom 𝑂 → (𝐻‘∪ dom 𝑂) = (𝑀‘∪ dom 𝑂))))
321	20, 320	mpcom 38	. . . . . 6 ⊢ (𝜑 → (∪ dom 𝑂 ∈ dom 𝑂 → (𝐻‘∪ dom 𝑂) = (𝑀‘∪ dom 𝑂)))
322	62, 321	mpd 15	. . . . 5 ⊢ (𝜑 → (𝐻‘∪ dom 𝑂) = (𝑀‘∪ dom 𝑂))
323	1, 2, 3, 220, 5, 232	cantnfval 9589	. . . . 5 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = (𝑀‘dom 𝐾))
324	228, 322, 323	3eqtr4d 2781	. . . 4 ⊢ (𝜑 → (𝐻‘∪ dom 𝑂) = ((𝐴 CNF 𝐵)‘𝐺))
325	140, 324	oveq12d 7385	. . 3 ⊢ (𝜑 → (((𝐴 ↑_o (𝑂‘∪ dom 𝑂)) ·_o (𝐹‘(𝑂‘∪ dom 𝑂))) +_o (𝐻‘∪ dom 𝑂)) = (((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o ((𝐴 CNF 𝐵)‘𝐺)))
326	22, 325	eqtrd 2771	. 2 ⊢ (𝜑 → (𝐻‘suc ∪ dom 𝑂) = (((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o ((𝐴 CNF 𝐵)‘𝐺)))
327	12, 15, 326	3eqtrd 2775	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o ((𝐴 CNF 𝐵)‘𝐺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 Vcvv 3429 ∖ cdif 3886 ∪ cun 3887 ∩ cin 3888 ⊆ wss 3889 ∅c0 4273 ifcif 4466 {csn 4567 ∪ cuni 4850 class class class wbr 5085 ↦ cmpt 5166 Tr wtr 5192 E cep 5530 Se wse 5582 We wwe 5583 ^◡ccnv 5630 dom cdm 5631 ↾ cres 5633 “ cima 5634 Ord word 6322 Oncon0 6323 suc csuc 6325 Fun wfun 6492 Fn wfn 6493 ⟶wf 6494 –onto→wfo 6496 –1-1-onto→wf1o 6497 ‘cfv 6498 Isom wiso 6499 (class class class)co 7367 ∈ cmpo 7369 ωcom 7817 supp csupp 8110 seq_ωcseqom 8386 1_oc1o 8398 +_o coa 8402 ·_o comu 8403 ↑_o coe 8404 finSupp cfsupp 9274 OrdIsocoi 9424 CNF ccnf 9582

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-seqom 8387 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-oi 9425 df-cnf 9583

This theorem is referenced by: cantnfp1 9602

W3C validator