Theorem cnfcom 9719

Description: Any ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)

Hypotheses

Ref	Expression
cnfcom.s	⊢ 𝑆 = dom (ω CNF 𝐴)
cnfcom.a	⊢ (𝜑 → 𝐴 ∈ On)
cnfcom.b	⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))
cnfcom.f	⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)
cnfcom.g	⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
cnfcom.h	⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)
cnfcom.t	⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
cnfcom.m	⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))
cnfcom.k	⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))
cnfcom.1	⊢ (𝜑 → 𝐼 ∈ dom 𝐺)

Assertion

Ref	Expression
cnfcom	⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))

Distinct variable groups:   𝑥,𝑘,𝑧,𝐴   𝑘,𝐼,𝑥,𝑧   𝑥,𝑀   𝑓,𝑘,𝑥,𝑧,𝐹   𝑧,𝑇   𝑓,𝐺,𝑘,𝑥,𝑧   𝑓,𝐻,𝑥   𝑆,𝑘,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑧,𝑓,𝑘)   𝐴(𝑓)   𝐵(𝑥,𝑧,𝑓,𝑘)   𝑆(𝑥,𝑓)   𝑇(𝑥,𝑓,𝑘)   𝐻(𝑧,𝑘)   𝐼(𝑓)   𝐾(𝑥,𝑧,𝑓,𝑘)   𝑀(𝑧,𝑓,𝑘)

Proof of Theorem cnfcom

Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnfcom.1	. 2 ⊢ (𝜑 → 𝐼 ∈ dom 𝐺)
2		cnfcom.s	. . . . . 6 ⊢ 𝑆 = dom (ω CNF 𝐴)
3		omelon 9665	. . . . . . 7 ⊢ ω ∈ On
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → ω ∈ On)
5		cnfcom.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ On)
6		cnfcom.g	. . . . . 6 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
7		cnfcom.f	. . . . . . 7 ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)
8	2, 4, 5	cantnff1o 9715	. . . . . . . . 9 ⊢ (𝜑 → (ω CNF 𝐴):𝑆–1-1-onto→(ω ↑o 𝐴))
9		f1ocnv 6835	. . . . . . . . 9 ⊢ ((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑o 𝐴) → ◡(ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto→𝑆)
10		f1of 6823	. . . . . . . . 9 ⊢ (◡(ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto→𝑆 → ◡(ω CNF 𝐴):(ω ↑o 𝐴)⟶𝑆)
11	8, 9, 10	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ◡(ω CNF 𝐴):(ω ↑o 𝐴)⟶𝑆)
12		cnfcom.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))
13	11, 12	ffvelcdmd 7080	. . . . . . 7 ⊢ (𝜑 → (◡(ω CNF 𝐴)‘𝐵) ∈ 𝑆)
14	7, 13	eqeltrid 2839	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
15	2, 4, 5, 6, 14	cantnfcl 9686	. . . . 5 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
16	15	simprd 495	. . . 4 ⊢ (𝜑 → dom 𝐺 ∈ ω)
17		elnn 7877	. . . 4 ⊢ ((𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω) → 𝐼 ∈ ω)
18	1, 16, 17	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐼 ∈ ω)
19		eleq1 2823	. . . . . 6 ⊢ (𝑤 = 𝐼 → (𝑤 ∈ dom 𝐺 ↔ 𝐼 ∈ dom 𝐺))
20		suceq 6424	. . . . . . . 8 ⊢ (𝑤 = 𝐼 → suc 𝑤 = suc 𝐼)
21	20	fveq2d 6885	. . . . . . 7 ⊢ (𝑤 = 𝐼 → (𝑇‘suc 𝑤) = (𝑇‘suc 𝐼))
22	20	fveq2d 6885	. . . . . . 7 ⊢ (𝑤 = 𝐼 → (𝐻‘suc 𝑤) = (𝐻‘suc 𝐼))
23		fveq2 6881	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → (𝐺‘𝑤) = (𝐺‘𝐼))
24	23	oveq2d 7426	. . . . . . . 8 ⊢ (𝑤 = 𝐼 → (ω ↑o (𝐺‘𝑤)) = (ω ↑o (𝐺‘𝐼)))
25		2fveq3 6886	. . . . . . . 8 ⊢ (𝑤 = 𝐼 → (𝐹‘(𝐺‘𝑤)) = (𝐹‘(𝐺‘𝐼)))
26	24, 25	oveq12d 7428	. . . . . . 7 ⊢ (𝑤 = 𝐼 → ((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤))) = ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))
27	21, 22, 26	f1oeq123d 6817	. . . . . 6 ⊢ (𝑤 = 𝐼 → ((𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤))) ↔ (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))))
28	19, 27	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝐼 → ((𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤)))) ↔ (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))))
29	28	imbi2d 340	. . . 4 ⊢ (𝑤 = 𝐼 → ((𝜑 → (𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤))))) ↔ (𝜑 → (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))))))
30		eleq1 2823	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ∈ dom 𝐺 ↔ ∅ ∈ dom 𝐺))
31		suceq 6424	. . . . . . . 8 ⊢ (𝑤 = ∅ → suc 𝑤 = suc ∅)
32	31	fveq2d 6885	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑇‘suc 𝑤) = (𝑇‘suc ∅))
33	31	fveq2d 6885	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝐻‘suc 𝑤) = (𝐻‘suc ∅))
34		fveq2 6881	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝐺‘𝑤) = (𝐺‘∅))
35	34	oveq2d 7426	. . . . . . . 8 ⊢ (𝑤 = ∅ → (ω ↑o (𝐺‘𝑤)) = (ω ↑o (𝐺‘∅)))
36		2fveq3 6886	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝐹‘(𝐺‘𝑤)) = (𝐹‘(𝐺‘∅)))
37	35, 36	oveq12d 7428	. . . . . . 7 ⊢ (𝑤 = ∅ → ((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤))) = ((ω ↑o (𝐺‘∅)) ·o (𝐹‘(𝐺‘∅))))
38	32, 33, 37	f1oeq123d 6817	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤))) ↔ (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑o (𝐺‘∅)) ·o (𝐹‘(𝐺‘∅)))))
39	30, 38	imbi12d 344	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤)))) ↔ (∅ ∈ dom 𝐺 → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑o (𝐺‘∅)) ·o (𝐹‘(𝐺‘∅))))))
40		eleq1 2823	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ dom 𝐺 ↔ 𝑦 ∈ dom 𝐺))
41		suceq 6424	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → suc 𝑤 = suc 𝑦)
42	41	fveq2d 6885	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑇‘suc 𝑤) = (𝑇‘suc 𝑦))
43	41	fveq2d 6885	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝐻‘suc 𝑤) = (𝐻‘suc 𝑦))
44		fveq2 6881	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐺‘𝑤) = (𝐺‘𝑦))
45	44	oveq2d 7426	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (ω ↑o (𝐺‘𝑤)) = (ω ↑o (𝐺‘𝑦)))
46		2fveq3 6886	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝐹‘(𝐺‘𝑤)) = (𝐹‘(𝐺‘𝑦)))
47	45, 46	oveq12d 7428	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤))) = ((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))
48	42, 43, 47	f1oeq123d 6817	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤))) ↔ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦)))))
49	40, 48	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤)))) ↔ (𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))))
50		eleq1 2823	. . . . . 6 ⊢ (𝑤 = suc 𝑦 → (𝑤 ∈ dom 𝐺 ↔ suc 𝑦 ∈ dom 𝐺))
51		suceq 6424	. . . . . . . 8 ⊢ (𝑤 = suc 𝑦 → suc 𝑤 = suc suc 𝑦)
52	51	fveq2d 6885	. . . . . . 7 ⊢ (𝑤 = suc 𝑦 → (𝑇‘suc 𝑤) = (𝑇‘suc suc 𝑦))
53	51	fveq2d 6885	. . . . . . 7 ⊢ (𝑤 = suc 𝑦 → (𝐻‘suc 𝑤) = (𝐻‘suc suc 𝑦))
54		fveq2 6881	. . . . . . . . 9 ⊢ (𝑤 = suc 𝑦 → (𝐺‘𝑤) = (𝐺‘suc 𝑦))
55	54	oveq2d 7426	. . . . . . . 8 ⊢ (𝑤 = suc 𝑦 → (ω ↑o (𝐺‘𝑤)) = (ω ↑o (𝐺‘suc 𝑦)))
56		2fveq3 6886	. . . . . . . 8 ⊢ (𝑤 = suc 𝑦 → (𝐹‘(𝐺‘𝑤)) = (𝐹‘(𝐺‘suc 𝑦)))
57	55, 56	oveq12d 7428	. . . . . . 7 ⊢ (𝑤 = suc 𝑦 → ((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤))) = ((ω ↑o (𝐺‘suc 𝑦)) ·o (𝐹‘(𝐺‘suc 𝑦))))
58	52, 53, 57	f1oeq123d 6817	. . . . . 6 ⊢ (𝑤 = suc 𝑦 → ((𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤))) ↔ (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘suc 𝑦)) ·o (𝐹‘(𝐺‘suc 𝑦)))))
59	50, 58	imbi12d 344	. . . . 5 ⊢ (𝑤 = suc 𝑦 → ((𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤)))) ↔ (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘suc 𝑦)) ·o (𝐹‘(𝐺‘suc 𝑦))))))
60	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → 𝐴 ∈ On)
61	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → 𝐵 ∈ (ω ↑o 𝐴))
62		cnfcom.h	. . . . . . 7 ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)
63		cnfcom.t	. . . . . . 7 ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
64		cnfcom.m	. . . . . . 7 ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))
65		cnfcom.k	. . . . . . 7 ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))
66		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ dom 𝐺)
67	3	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ω ∈ On)
68		suppssdm 8181	. . . . . . . . . . 11 ⊢ (𝐹 supp ∅) ⊆ dom 𝐹
69	2, 4, 5	cantnfs 9685	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
70	14, 69	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
71	70	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶ω)
72	68, 71	fssdm 6730	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
73		onss 7784	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
74	5, 73	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ On)
75	72, 74	sstrd 3974	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ On)
76	6	oif 9549	. . . . . . . . . 10 ⊢ 𝐺:dom 𝐺⟶(𝐹 supp ∅)
77	76	ffvelcdmi 7078	. . . . . . . . 9 ⊢ (∅ ∈ dom 𝐺 → (𝐺‘∅) ∈ (𝐹 supp ∅))
78		ssel2 3958	. . . . . . . . 9 ⊢ (((𝐹 supp ∅) ⊆ On ∧ (𝐺‘∅) ∈ (𝐹 supp ∅)) → (𝐺‘∅) ∈ On)
79	75, 77, 78	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ On)
80		peano1 7889	. . . . . . . . 9 ⊢ ∅ ∈ ω
81	80	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ ω)
82		oen0 8603	. . . . . . . 8 ⊢ (((ω ∈ On ∧ (𝐺‘∅) ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o (𝐺‘∅)))
83	67, 79, 81, 82	syl21anc 837	. . . . . . 7 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ (ω ↑o (𝐺‘∅)))
84		0ex 5282	. . . . . . . . 9 ⊢ ∅ ∈ V
85	63	seqom0g 8475	. . . . . . . . 9 ⊢ (∅ ∈ V → (𝑇‘∅) = ∅)
86	84, 85	ax-mp 5	. . . . . . . 8 ⊢ (𝑇‘∅) = ∅
87		f1o0 6860	. . . . . . . . . 10 ⊢ ∅:∅–1-1-onto→∅
88	62	seqom0g 8475	. . . . . . . . . . 11 ⊢ (∅ ∈ V → (𝐻‘∅) = ∅)
89		f1oeq2 6812	. . . . . . . . . . 11 ⊢ ((𝐻‘∅) = ∅ → (∅:(𝐻‘∅)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
90	84, 88, 89	mp2b 10	. . . . . . . . . 10 ⊢ (∅:(𝐻‘∅)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)
91	87, 90	mpbir 231	. . . . . . . . 9 ⊢ ∅:(𝐻‘∅)–1-1-onto→∅
92		f1oeq1 6811	. . . . . . . . 9 ⊢ ((𝑇‘∅) = ∅ → ((𝑇‘∅):(𝐻‘∅)–1-1-onto→∅ ↔ ∅:(𝐻‘∅)–1-1-onto→∅))
93	91, 92	mpbiri 258	. . . . . . . 8 ⊢ ((𝑇‘∅) = ∅ → (𝑇‘∅):(𝐻‘∅)–1-1-onto→∅)
94	86, 93	mp1i 13	. . . . . . 7 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝑇‘∅):(𝐻‘∅)–1-1-onto→∅)
95	2, 60, 61, 7, 6, 62, 63, 64, 65, 66, 83, 94	cnfcomlem 9718	. . . . . 6 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑o (𝐺‘∅)) ·o (𝐹‘(𝐺‘∅))))
96	95	ex 412	. . . . 5 ⊢ (𝜑 → (∅ ∈ dom 𝐺 → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑o (𝐺‘∅)) ·o (𝐹‘(𝐺‘∅)))))
97	6	oicl 9548	. . . . . . . . . 10 ⊢ Ord dom 𝐺
98		ordtr 6371	. . . . . . . . . 10 ⊢ (Ord dom 𝐺 → Tr dom 𝐺)
99	97, 98	ax-mp 5	. . . . . . . . 9 ⊢ Tr dom 𝐺
100		trsuc 6446	. . . . . . . . 9 ⊢ ((Tr dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺) → 𝑦 ∈ dom 𝐺)
101	99, 100	mpan 690	. . . . . . . 8 ⊢ (suc 𝑦 ∈ dom 𝐺 → 𝑦 ∈ dom 𝐺)
102	101	imim1i 63	. . . . . . 7 ⊢ ((𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦)))) → (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦)))))
103	5	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → 𝐴 ∈ On)
104	12	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → 𝐵 ∈ (ω ↑o 𝐴))
105		simprl 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → suc 𝑦 ∈ dom 𝐺)
106	74	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → 𝐴 ⊆ On)
107	72	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝐹 supp ∅) ⊆ 𝐴)
108	76	ffvelcdmi 7078	. . . . . . . . . . . . . . . . 17 ⊢ (suc 𝑦 ∈ dom 𝐺 → (𝐺‘suc 𝑦) ∈ (𝐹 supp ∅))
109	108	ad2antrl 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝐺‘suc 𝑦) ∈ (𝐹 supp ∅))
110	107, 109	sseldd 3964	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝐺‘suc 𝑦) ∈ 𝐴)
111	106, 110	sseldd 3964	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝐺‘suc 𝑦) ∈ On)
112		eloni 6367	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘suc 𝑦) ∈ On → Ord (𝐺‘suc 𝑦))
113	111, 112	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → Ord (𝐺‘suc 𝑦))
114		vex 3468	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
115	114	sucid 6441	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ suc 𝑦
116		ovexd 7445	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V)
117	15	simpld 494	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → E We (𝐹 supp ∅))
118	6	oiiso 9556	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
119	116, 117, 118	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
120	119	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
121	101	ad2antrl 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → 𝑦 ∈ dom 𝐺)
122		isorel 7324	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) ∧ (𝑦 ∈ dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺)) → (𝑦 E suc 𝑦 ↔ (𝐺‘𝑦) E (𝐺‘suc 𝑦)))
123	120, 121, 105, 122	syl12anc 836	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝑦 E suc 𝑦 ↔ (𝐺‘𝑦) E (𝐺‘suc 𝑦)))
124	114	sucex 7805	. . . . . . . . . . . . . . . 16 ⊢ suc 𝑦 ∈ V
125	124	epeli 5560	. . . . . . . . . . . . . . 15 ⊢ (𝑦 E suc 𝑦 ↔ 𝑦 ∈ suc 𝑦)
126		fvex 6894	. . . . . . . . . . . . . . . 16 ⊢ (𝐺‘suc 𝑦) ∈ V
127	126	epeli 5560	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑦) E (𝐺‘suc 𝑦) ↔ (𝐺‘𝑦) ∈ (𝐺‘suc 𝑦))
128	123, 125, 127	3bitr3g 313	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝑦 ∈ suc 𝑦 ↔ (𝐺‘𝑦) ∈ (𝐺‘suc 𝑦)))
129	115, 128	mpbii 233	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝐺‘𝑦) ∈ (𝐺‘suc 𝑦))
130		ordsucss 7817	. . . . . . . . . . . . 13 ⊢ (Ord (𝐺‘suc 𝑦) → ((𝐺‘𝑦) ∈ (𝐺‘suc 𝑦) → suc (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)))
131	113, 129, 130	sylc 65	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → suc (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦))
132	76	ffvelcdmi 7078	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ dom 𝐺 → (𝐺‘𝑦) ∈ (𝐹 supp ∅))
133	121, 132	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝐺‘𝑦) ∈ (𝐹 supp ∅))
134	107, 133	sseldd 3964	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝐺‘𝑦) ∈ 𝐴)
135	106, 134	sseldd 3964	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝐺‘𝑦) ∈ On)
136		onsuc 7810	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘𝑦) ∈ On → suc (𝐺‘𝑦) ∈ On)
137	135, 136	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → suc (𝐺‘𝑦) ∈ On)
138	3	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → ω ∈ On)
139	80	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → ∅ ∈ ω)
140		oewordi 8608	. . . . . . . . . . . . 13 ⊢ (((suc (𝐺‘𝑦) ∈ On ∧ (𝐺‘suc 𝑦) ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → (suc (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → (ω ↑o suc (𝐺‘𝑦)) ⊆ (ω ↑o (𝐺‘suc 𝑦))))
141	137, 111, 138, 139, 140	syl31anc 1375	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (suc (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → (ω ↑o suc (𝐺‘𝑦)) ⊆ (ω ↑o (𝐺‘suc 𝑦))))
142	131, 141	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (ω ↑o suc (𝐺‘𝑦)) ⊆ (ω ↑o (𝐺‘suc 𝑦)))
143	71	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → 𝐹:𝐴⟶ω)
144	143, 134	ffvelcdmd 7080	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝐹‘(𝐺‘𝑦)) ∈ ω)
145		nnon 7872	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘(𝐺‘𝑦)) ∈ ω → (𝐹‘(𝐺‘𝑦)) ∈ On)
146	144, 145	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝐹‘(𝐺‘𝑦)) ∈ On)
147		oecl 8554	. . . . . . . . . . . . . . 15 ⊢ ((ω ∈ On ∧ (𝐺‘𝑦) ∈ On) → (ω ↑o (𝐺‘𝑦)) ∈ On)
148	138, 135, 147	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (ω ↑o (𝐺‘𝑦)) ∈ On)
149		oen0 8603	. . . . . . . . . . . . . . 15 ⊢ (((ω ∈ On ∧ (𝐺‘𝑦) ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o (𝐺‘𝑦)))
150	138, 135, 139, 149	syl21anc 837	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → ∅ ∈ (ω ↑o (𝐺‘𝑦)))
151		omord2 8584	. . . . . . . . . . . . . 14 ⊢ ((((𝐹‘(𝐺‘𝑦)) ∈ On ∧ ω ∈ On ∧ (ω ↑o (𝐺‘𝑦)) ∈ On) ∧ ∅ ∈ (ω ↑o (𝐺‘𝑦))) → ((𝐹‘(𝐺‘𝑦)) ∈ ω ↔ ((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))) ∈ ((ω ↑o (𝐺‘𝑦)) ·o ω)))
152	146, 138, 148, 150, 151	syl31anc 1375	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → ((𝐹‘(𝐺‘𝑦)) ∈ ω ↔ ((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))) ∈ ((ω ↑o (𝐺‘𝑦)) ·o ω)))
153	144, 152	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → ((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))) ∈ ((ω ↑o (𝐺‘𝑦)) ·o ω))
154		oesuc 8544	. . . . . . . . . . . . 13 ⊢ ((ω ∈ On ∧ (𝐺‘𝑦) ∈ On) → (ω ↑o suc (𝐺‘𝑦)) = ((ω ↑o (𝐺‘𝑦)) ·o ω))
155	138, 135, 154	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (ω ↑o suc (𝐺‘𝑦)) = ((ω ↑o (𝐺‘𝑦)) ·o ω))
156	153, 155	eleqtrrd 2838	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → ((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))) ∈ (ω ↑o suc (𝐺‘𝑦)))
157	142, 156	sseldd 3964	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → ((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))) ∈ (ω ↑o (𝐺‘suc 𝑦)))
158		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))
159	2, 103, 104, 7, 6, 62, 63, 64, 65, 105, 157, 158	cnfcomlem 9718	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))))) → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘suc 𝑦)) ·o (𝐹‘(𝐺‘suc 𝑦))))
160	159	exp32 420	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (suc 𝑦 ∈ dom 𝐺 → ((𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦))) → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘suc 𝑦)) ·o (𝐹‘(𝐺‘suc 𝑦))))))
161	160	a2d 29	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦)))) → (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘suc 𝑦)) ·o (𝐹‘(𝐺‘suc 𝑦))))))
162	102, 161	syl5 34	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦)))) → (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘suc 𝑦)) ·o (𝐹‘(𝐺‘suc 𝑦))))))
163	162	expcom 413	. . . . 5 ⊢ (𝑦 ∈ ω → (𝜑 → ((𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘𝑦)) ·o (𝐹‘(𝐺‘𝑦)))) → (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑o (𝐺‘suc 𝑦)) ·o (𝐹‘(𝐺‘suc 𝑦)))))))
164	39, 49, 59, 96, 163	finds2 7899	. . . 4 ⊢ (𝑤 ∈ ω → (𝜑 → (𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑o (𝐺‘𝑤)) ·o (𝐹‘(𝐺‘𝑤))))))
165	29, 164	vtoclga 3561	. . 3 ⊢ (𝐼 ∈ ω → (𝜑 → (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))))
166	18, 165	mpcom 38	. 2 ⊢ (𝜑 → (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))))
167	1, 166	mpd 15	1 ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∪ cun 3929   ⊆ wss 3931  ∅c0 4313   class class class wbr 5124   ↦ cmpt 5206  Tr wtr 5234   E cep 5557   We wwe 5610  ^◡ccnv 5658  dom cdm 5659  Ord word 6356  Oncon0 6357  suc csuc 6359  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536   Isom wiso 6537  (class class class)co 7410   ∈ cmpo 7412  ωcom 7866   supp csupp 8164  seq_ωcseqom 8466   +_o coa 8482   ·_o comu 8483   ↑_o coe 8484   finSupp cfsupp 9378  OrdIsocoi 9528   CNF ccnf 9680

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-supp 8165  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-seqom 8467  df-1o 8485  df-2o 8486  df-oadd 8489  df-omul 8490  df-oexp 8491  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9379  df-oi 9529  df-cnf 9681

This theorem is referenced by:  cnfcom2  9721