Theorem cnfcomlem 9008

Description: Lemma for cnfcom 9009. (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)
cnfcom.2	⊢ (𝜑 → 𝑂 ∈ (ω ↑o (𝐺‘𝐼)))
cnfcom.3	⊢ (𝜑 → (𝑇‘𝐼):(𝐻‘𝐼)–1-1-onto→𝑂)

Assertion

Ref	Expression
cnfcomlem	⊢ (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)   𝑂(𝑥,𝑧,𝑓,𝑘)

Proof of Theorem cnfcomlem

Dummy variables 𝑢 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omelon 8955	. . . . . . 7 ⊢ ω ∈ On
2		cnfcom.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ On)
3		suppssdm 7694	. . . . . . . . . 10 ⊢ (𝐹 supp ∅) ⊆ dom 𝐹
4		cnfcom.f	. . . . . . . . . . . . 13 ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)
5		cnfcom.s	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 = dom (ω CNF 𝐴)
6	1	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ω ∈ On)
7	5, 6, 2	cantnff1o 9005	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ω CNF 𝐴):𝑆–1-1-onto→(ω ↑o 𝐴))
8		f1ocnv 6495	. . . . . . . . . . . . . . 15 ⊢ ((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑o 𝐴) → ◡(ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto→𝑆)
9		f1of 6483	. . . . . . . . . . . . . . 15 ⊢ (◡(ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto→𝑆 → ◡(ω CNF 𝐴):(ω ↑o 𝐴)⟶𝑆)
10	7, 8, 9	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ◡(ω CNF 𝐴):(ω ↑o 𝐴)⟶𝑆)
11		cnfcom.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))
12	10, 11	ffvelrnd 6717	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡(ω CNF 𝐴)‘𝐵) ∈ 𝑆)
13	4, 12	syl5eqel 2887	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
14	5, 6, 2	cantnfs 8975	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
15	13, 14	mpbid 233	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
16	15	simpld 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶ω)
17	3, 16	fssdm 6398	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
18		cnfcom.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ dom 𝐺)
19		cnfcom.g	. . . . . . . . . . . 12 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
20	19	oif 8840	. . . . . . . . . . 11 ⊢ 𝐺:dom 𝐺⟶(𝐹 supp ∅)
21	20	ffvelrni 6715	. . . . . . . . . 10 ⊢ (𝐼 ∈ dom 𝐺 → (𝐺‘𝐼) ∈ (𝐹 supp ∅))
22	18, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝐼) ∈ (𝐹 supp ∅))
23	17, 22	sseldd 3890	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐼) ∈ 𝐴)
24		onelon 6091	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝐼) ∈ 𝐴) → (𝐺‘𝐼) ∈ On)
25	2, 23, 24	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝐼) ∈ On)
26		oecl 8013	. . . . . . 7 ⊢ ((ω ∈ On ∧ (𝐺‘𝐼) ∈ On) → (ω ↑o (𝐺‘𝐼)) ∈ On)
27	1, 25, 26	sylancr 587	. . . . . 6 ⊢ (𝜑 → (ω ↑o (𝐺‘𝐼)) ∈ On)
28	16, 23	ffvelrnd 6717	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝐺‘𝐼)) ∈ ω)
29		nnon 7442	. . . . . . 7 ⊢ ((𝐹‘(𝐺‘𝐼)) ∈ ω → (𝐹‘(𝐺‘𝐼)) ∈ On)
30	28, 29	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐺‘𝐼)) ∈ On)
31		omcl 8012	. . . . . 6 ⊢ (((ω ↑o (𝐺‘𝐼)) ∈ On ∧ (𝐹‘(𝐺‘𝐼)) ∈ On) → ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ∈ On)
32	27, 30, 31	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ∈ On)
33	5, 6, 2, 19, 13	cantnfcl 8976	. . . . . . . 8 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
34	33	simprd 496	. . . . . . 7 ⊢ (𝜑 → dom 𝐺 ∈ ω)
35		elnn 7446	. . . . . . 7 ⊢ ((𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω) → 𝐼 ∈ ω)
36	18, 34, 35	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ω)
37		cnfcom.h	. . . . . . . 8 ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)
38	37	cantnfvalf 8974	. . . . . . 7 ⊢ 𝐻:ω⟶On
39	38	ffvelrni 6715	. . . . . 6 ⊢ (𝐼 ∈ ω → (𝐻‘𝐼) ∈ On)
40	36, 39	syl 17	. . . . 5 ⊢ (𝜑 → (𝐻‘𝐼) ∈ On)
41		eqid 2795	. . . . . 6 ⊢ ((𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∪ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦))) = ((𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∪ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦)))
42	41	oacomf1o 8041	. . . . 5 ⊢ ((((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ∈ On ∧ (𝐻‘𝐼) ∈ On) → ((𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∪ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦))):(((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o (𝐻‘𝐼))–1-1-onto→((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))))
43	32, 40, 42	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∪ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦))):(((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o (𝐻‘𝐼))–1-1-onto→((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))))
44		cnfcom.t	. . . . . . . 8 ⊢ 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
45	44	seqomsuc 7944	. . . . . . 7 ⊢ (𝐼 ∈ ω → (𝑇‘suc 𝐼) = (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇‘𝐼)))
46	36, 45	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑇‘suc 𝐼) = (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇‘𝐼)))
47		nfcv 2949	. . . . . . . . 9 ⊢ Ⅎ𝑢𝐾
48		nfcv 2949	. . . . . . . . 9 ⊢ Ⅎ𝑣𝐾
49		nfcv 2949	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝑦 ∈ ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) ↦ (dom 𝑣 +o 𝑦)) ∪ ◡(𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦)))
50		nfcv 2949	. . . . . . . . 9 ⊢ Ⅎ𝑓((𝑦 ∈ ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) ↦ (dom 𝑣 +o 𝑦)) ∪ ◡(𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦)))
51		cnfcom.k	. . . . . . . . . 10 ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))
52		oveq2 7024	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (dom 𝑓 +o 𝑥) = (dom 𝑓 +o 𝑦))
53	52	cbvmptv 5061	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) = (𝑦 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑦))
54		cnfcom.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))
55		simpl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → 𝑘 = 𝑢)
56	55	fveq2d 6542	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝐺‘𝑘) = (𝐺‘𝑢))
57	56	oveq2d 7032	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (ω ↑o (𝐺‘𝑘)) = (ω ↑o (𝐺‘𝑢)))
58	56	fveq2d 6542	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝐹‘(𝐺‘𝑘)) = (𝐹‘(𝐺‘𝑢)))
59	57, 58	oveq12d 7034	. . . . . . . . . . . . . 14 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) = ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))))
60	54, 59	syl5eq 2843	. . . . . . . . . . . . 13 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → 𝑀 = ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))))
61		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → 𝑓 = 𝑣)
62	61	dmeqd 5660	. . . . . . . . . . . . . 14 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → dom 𝑓 = dom 𝑣)
63	62	oveq1d 7031	. . . . . . . . . . . . 13 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (dom 𝑓 +o 𝑦) = (dom 𝑣 +o 𝑦))
64	60, 63	mpteq12dv 5045	. . . . . . . . . . . 12 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝑦 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑦)) = (𝑦 ∈ ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) ↦ (dom 𝑣 +o 𝑦)))
65	53, 64	syl5eq 2843	. . . . . . . . . . 11 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) = (𝑦 ∈ ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) ↦ (dom 𝑣 +o 𝑦)))
66		oveq2 7024	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑀 +o 𝑥) = (𝑀 +o 𝑦))
67	66	cbvmptv 5061	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)) = (𝑦 ∈ dom 𝑓 ↦ (𝑀 +o 𝑦))
68	60	oveq1d 7031	. . . . . . . . . . . . . 14 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝑀 +o 𝑦) = (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦))
69	62, 68	mpteq12dv 5045	. . . . . . . . . . . . 13 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝑦 ∈ dom 𝑓 ↦ (𝑀 +o 𝑦)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦)))
70	67, 69	syl5eq 2843	. . . . . . . . . . . 12 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦)))
71	70	cnveqd 5632	. . . . . . . . . . 11 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)) = ◡(𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦)))
72	65, 71	uneq12d 4061	. . . . . . . . . 10 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) = ((𝑦 ∈ ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) ↦ (dom 𝑣 +o 𝑦)) ∪ ◡(𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦))))
73	51, 72	syl5eq 2843	. . . . . . . . 9 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → 𝐾 = ((𝑦 ∈ ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) ↦ (dom 𝑣 +o 𝑦)) ∪ ◡(𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦))))
74	47, 48, 49, 50, 73	cbvmpo 7104	. . . . . . . 8 ⊢ (𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾) = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑦 ∈ ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) ↦ (dom 𝑣 +o 𝑦)) ∪ ◡(𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦))))
75	74	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾) = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑦 ∈ ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) ↦ (dom 𝑣 +o 𝑦)) ∪ ◡(𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦)))))
76		simprl 767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → 𝑢 = 𝐼)
77	76	fveq2d 6542	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → (𝐺‘𝑢) = (𝐺‘𝐼))
78	77	oveq2d 7032	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → (ω ↑o (𝐺‘𝑢)) = (ω ↑o (𝐺‘𝐼)))
79	77	fveq2d 6542	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → (𝐹‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝐼)))
80	78, 79	oveq12d 7034	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) = ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))
81		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼)) → 𝑣 = (𝑇‘𝐼))
82	81	dmeqd 5660	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼)) → dom 𝑣 = dom (𝑇‘𝐼))
83		cnfcom.3	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇‘𝐼):(𝐻‘𝐼)–1-1-onto→𝑂)
84		f1odm 6487	. . . . . . . . . . . 12 ⊢ ((𝑇‘𝐼):(𝐻‘𝐼)–1-1-onto→𝑂 → dom (𝑇‘𝐼) = (𝐻‘𝐼))
85	83, 84	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑇‘𝐼) = (𝐻‘𝐼))
86	82, 85	sylan9eqr 2853	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → dom 𝑣 = (𝐻‘𝐼))
87	86	oveq1d 7031	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → (dom 𝑣 +o 𝑦) = ((𝐻‘𝐼) +o 𝑦))
88	80, 87	mpteq12dv 5045	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → (𝑦 ∈ ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) ↦ (dom 𝑣 +o 𝑦)) = (𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)))
89	80	oveq1d 7031	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦) = (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦))
90	86, 89	mpteq12dv 5045	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → (𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦)) = (𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦)))
91	90	cnveqd 5632	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → ◡(𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦)) = ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦)))
92	88, 91	uneq12d 4061	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → ((𝑦 ∈ ((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) ↦ (dom 𝑣 +o 𝑦)) ∪ ◡(𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑦))) = ((𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∪ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦))))
93	18	elexd 3457	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V)
94		fvexd 6553	. . . . . . 7 ⊢ (𝜑 → (𝑇‘𝐼) ∈ V)
95		ovex 7048	. . . . . . . . . 10 ⊢ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ∈ V
96	95	mptex 6852	. . . . . . . . 9 ⊢ (𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∈ V
97		fvex 6551	. . . . . . . . . . 11 ⊢ (𝐻‘𝐼) ∈ V
98	97	mptex 6852	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦)) ∈ V
99	98	cnvex 7486	. . . . . . . . 9 ⊢ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦)) ∈ V
100	96, 99	unex 7326	. . . . . . . 8 ⊢ ((𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∪ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦))) ∈ V
101	100	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∪ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦))) ∈ V)
102	75, 92, 93, 94, 101	ovmpod 7158	. . . . . 6 ⊢ (𝜑 → (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇‘𝐼)) = ((𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∪ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦))))
103	46, 102	eqtrd 2831	. . . . 5 ⊢ (𝜑 → (𝑇‘suc 𝐼) = ((𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∪ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦))))
104		f1oeq1 6472	. . . . 5 ⊢ ((𝑇‘suc 𝐼) = ((𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∪ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦))) → ((𝑇‘suc 𝐼):(((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o (𝐻‘𝐼))–1-1-onto→((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))) ↔ ((𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∪ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦))):(((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o (𝐻‘𝐼))–1-1-onto→((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))))
105	103, 104	syl 17	. . . 4 ⊢ (𝜑 → ((𝑇‘suc 𝐼):(((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o (𝐻‘𝐼))–1-1-onto→((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))) ↔ ((𝑦 ∈ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +o 𝑦)) ∪ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o 𝑦))):(((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o (𝐻‘𝐼))–1-1-onto→((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))))
106	43, 105	mpbird 258	. . 3 ⊢ (𝜑 → (𝑇‘suc 𝐼):(((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o (𝐻‘𝐼))–1-1-onto→((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))))
107	1	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) → ω ∈ On)
108		simpl 483	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) → 𝐴 ∈ On)
109		simpr 485	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) → 𝐹 ∈ 𝑆)
110	54	oveq1i 7026	. . . . . . . . . 10 ⊢ (𝑀 +o 𝑧) = (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)
111	110	a1i 11	. . . . . . . . 9 ⊢ ((𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑀 +o 𝑧) = (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧))
112	111	mpoeq3ia 7090	. . . . . . . 8 ⊢ (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧))
113		eqid 2795	. . . . . . . 8 ⊢ ∅ = ∅
114		seqomeq12 7941	. . . . . . . 8 ⊢ (((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)) ∧ ∅ = ∅) → seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅))
115	112, 113, 114	mp2an 688	. . . . . . 7 ⊢ seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)
116	37, 115	eqtri 2819	. . . . . 6 ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)
117	5, 107, 108, 19, 109, 116	cantnfsuc 8979	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ 𝐼 ∈ ω) → (𝐻‘suc 𝐼) = (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o (𝐻‘𝐼)))
118	2, 13, 36, 117	syl21anc 834	. . . 4 ⊢ (𝜑 → (𝐻‘suc 𝐼) = (((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o (𝐻‘𝐼)))
119	118	f1oeq2d 6479	. . 3 ⊢ (𝜑 → ((𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))) ↔ (𝑇‘suc 𝐼):(((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) +o (𝐻‘𝐼))–1-1-onto→((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))))
120	106, 119	mpbird 258	. 2 ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))))
121		sssucid 6143	. . . . . 6 ⊢ dom 𝐺 ⊆ suc dom 𝐺
122	121, 18	sseldi 3887	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ suc dom 𝐺)
123		epelg 5354	. . . . . . . . . . 11 ⊢ (𝐼 ∈ dom 𝐺 → (𝑦 E 𝐼 ↔ 𝑦 ∈ 𝐼))
124	18, 123	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 E 𝐼 ↔ 𝑦 ∈ 𝐼))
125	124	biimpar 478	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝑦 E 𝐼)
126		ovexd 7050	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V)
127	33	simpld 495	. . . . . . . . . . . 12 ⊢ (𝜑 → E We (𝐹 supp ∅))
128	19	oiiso 8847	. . . . . . . . . . . 12 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
129	126, 127, 128	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
130	129	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
131	19	oicl 8839	. . . . . . . . . . . 12 ⊢ Ord dom 𝐺
132		ordelss 6082	. . . . . . . . . . . 12 ⊢ ((Ord dom 𝐺 ∧ 𝐼 ∈ dom 𝐺) → 𝐼 ⊆ dom 𝐺)
133	131, 18, 132	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ dom 𝐺)
134	133	sselda 3889	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ dom 𝐺)
135	18	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ dom 𝐺)
136		isorel 6942	. . . . . . . . . 10 ⊢ ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) ∧ (𝑦 ∈ dom 𝐺 ∧ 𝐼 ∈ dom 𝐺)) → (𝑦 E 𝐼 ↔ (𝐺‘𝑦) E (𝐺‘𝐼)))
137	130, 134, 135, 136	syl12anc 833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑦 E 𝐼 ↔ (𝐺‘𝑦) E (𝐺‘𝐼)))
138	125, 137	mpbid 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐺‘𝑦) E (𝐺‘𝐼))
139		fvex 6551	. . . . . . . . 9 ⊢ (𝐺‘𝐼) ∈ V
140	139	epeli 5356	. . . . . . . 8 ⊢ ((𝐺‘𝑦) E (𝐺‘𝐼) ↔ (𝐺‘𝑦) ∈ (𝐺‘𝐼))
141	138, 140	sylib 219	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐺‘𝑦) ∈ (𝐺‘𝐼))
142	141	ralrimiva 3149	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 (𝐺‘𝑦) ∈ (𝐺‘𝐼))
143		ffun 6385	. . . . . . . 8 ⊢ (𝐺:dom 𝐺⟶(𝐹 supp ∅) → Fun 𝐺)
144	20, 143	ax-mp 5	. . . . . . 7 ⊢ Fun 𝐺
145		funimass4 6598	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐼 ⊆ dom 𝐺) → ((𝐺 “ 𝐼) ⊆ (𝐺‘𝐼) ↔ ∀𝑦 ∈ 𝐼 (𝐺‘𝑦) ∈ (𝐺‘𝐼)))
146	144, 133, 145	sylancr 587	. . . . . 6 ⊢ (𝜑 → ((𝐺 “ 𝐼) ⊆ (𝐺‘𝐼) ↔ ∀𝑦 ∈ 𝐼 (𝐺‘𝑦) ∈ (𝐺‘𝐼)))
147	142, 146	mpbird 258	. . . . 5 ⊢ (𝜑 → (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))
148	1	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → ω ∈ On)
149		simpll 763	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → 𝐴 ∈ On)
150		simplr 765	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → 𝐹 ∈ 𝑆)
151		peano1 7457	. . . . . . 7 ⊢ ∅ ∈ ω
152	151	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → ∅ ∈ ω)
153		simpr1 1187	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → 𝐼 ∈ suc dom 𝐺)
154		simpr2 1188	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → (𝐺‘𝐼) ∈ On)
155		simpr3 1189	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))
156	5, 148, 149, 19, 150, 116, 152, 153, 154, 155	cantnflt 8981	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → (𝐻‘𝐼) ∈ (ω ↑o (𝐺‘𝐼)))
157	2, 13, 122, 25, 147, 156	syl23anc 1370	. . . 4 ⊢ (𝜑 → (𝐻‘𝐼) ∈ (ω ↑o (𝐺‘𝐼)))
158	16	ffnd 6383	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐴)
159		0ex 5102	. . . . . . . . . 10 ⊢ ∅ ∈ V
160	159	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ V)
161		elsuppfn 7689	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ On ∧ ∅ ∈ V) → ((𝐺‘𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺‘𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝐼)) ≠ ∅)))
162	158, 2, 160, 161	syl3anc 1364	. . . . . . . 8 ⊢ (𝜑 → ((𝐺‘𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺‘𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝐼)) ≠ ∅)))
163		simpr 485	. . . . . . . 8 ⊢ (((𝐺‘𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝐼)) ≠ ∅) → (𝐹‘(𝐺‘𝐼)) ≠ ∅)
164	162, 163	syl6bi 254	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝐼) ∈ (𝐹 supp ∅) → (𝐹‘(𝐺‘𝐼)) ≠ ∅))
165	22, 164	mpd 15	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐺‘𝐼)) ≠ ∅)
166		on0eln0 6121	. . . . . . 7 ⊢ ((𝐹‘(𝐺‘𝐼)) ∈ On → (∅ ∈ (𝐹‘(𝐺‘𝐼)) ↔ (𝐹‘(𝐺‘𝐼)) ≠ ∅))
167	30, 166	syl 17	. . . . . 6 ⊢ (𝜑 → (∅ ∈ (𝐹‘(𝐺‘𝐼)) ↔ (𝐹‘(𝐺‘𝐼)) ≠ ∅))
168	165, 167	mpbird 258	. . . . 5 ⊢ (𝜑 → ∅ ∈ (𝐹‘(𝐺‘𝐼)))
169		omword1 8049	. . . . 5 ⊢ ((((ω ↑o (𝐺‘𝐼)) ∈ On ∧ (𝐹‘(𝐺‘𝐼)) ∈ On) ∧ ∅ ∈ (𝐹‘(𝐺‘𝐼))) → (ω ↑o (𝐺‘𝐼)) ⊆ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))
170	27, 30, 168, 169	syl21anc 834	. . . 4 ⊢ (𝜑 → (ω ↑o (𝐺‘𝐼)) ⊆ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))
171		oaabs2 8122	. . . 4 ⊢ ((((𝐻‘𝐼) ∈ (ω ↑o (𝐺‘𝐼)) ∧ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))) ∈ On) ∧ (ω ↑o (𝐺‘𝐼)) ⊆ ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))) → ((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))) = ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))
172	157, 32, 170, 171	syl21anc 834	. . 3 ⊢ (𝜑 → ((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))) = ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))
173	172	f1oeq3d 6480	. 2 ⊢ (𝜑 → ((𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻‘𝐼) +o ((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))) ↔ (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))))
174	120, 173	mpbid 233	1 ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081   ≠ wne 2984  ∀wral 3105  Vcvv 3437   ∪ cun 3857   ⊆ wss 3859  ∅c0 4211   class class class wbr 4962   ↦ cmpt 5041   E cep 5352   We wwe 5401  ^◡ccnv 5442  dom cdm 5443   “ cima 5446  Ord word 6065  Oncon0 6066  suc csuc 6068  Fun wfun 6219   Fn wfn 6220  ⟶wf 6221  –1-1-onto→wf1o 6224  ‘cfv 6225   Isom wiso 6226  (class class class)co 7016   ∈ cmpo 7018  ωcom 7436   supp csupp 7681  seq_𝜔cseqom 7934   +_o coa 7950   ·_o comu 7951   ↑_o coe 7952   finSupp cfsupp 8679  OrdIsocoi 8819   CNF ccnf 8970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-supp 7682  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-seqom 7935  df-1o 7953  df-2o 7954  df-oadd 7957  df-omul 7958  df-oexp 7959  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fsupp 8680  df-oi 8820  df-cnf 8971

This theorem is referenced by:  cnfcom  9009