Theorem cnfcomlem 8596

Description: Lemma for cnfcom 8597. (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 8543	. . . . . . 7 ⊢ ω ∈ On
2		cnfcom.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ On)
3		suppssdm 7308	. . . . . . . . . 10 ⊢ (𝐹 supp ∅) ⊆ dom 𝐹
4		cnfcom.f	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)
5		cnfcom.s	. . . . . . . . . . . . . . . . 17 ⊢ 𝑆 = dom (ω CNF 𝐴)
6	1	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ω ∈ On)
7	5, 6, 2	cantnff1o 8593	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ω CNF 𝐴):𝑆–1-1-onto→(ω ↑𝑜 𝐴))
8		f1ocnv 6149	. . . . . . . . . . . . . . . 16 ⊢ ((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑𝑜 𝐴) → ◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto→𝑆)
9		f1of 6137	. . . . . . . . . . . . . . . 16 ⊢ (◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto→𝑆 → ◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
10	7, 8, 9	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
11		cnfcom.b	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ (ω ↑𝑜 𝐴))
12	10, 11	ffvelrnd 6360	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡(ω CNF 𝐴)‘𝐵) ∈ 𝑆)
13	4, 12	syl5eqel 2705	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
14	5, 6, 2	cantnfs 8563	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
15	13, 14	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
16	15	simpld 475	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶ω)
17		fdm 6051	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ω → dom 𝐹 = 𝐴)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝐴)
19	3, 18	syl5sseq 3653	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
20		cnfcom.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ dom 𝐺)
21		cnfcom.g	. . . . . . . . . . . 12 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
22	21	oif 8435	. . . . . . . . . . 11 ⊢ 𝐺:dom 𝐺⟶(𝐹 supp ∅)
23	22	ffvelrni 6358	. . . . . . . . . 10 ⊢ (𝐼 ∈ dom 𝐺 → (𝐺‘𝐼) ∈ (𝐹 supp ∅))
24	20, 23	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝐼) ∈ (𝐹 supp ∅))
25	19, 24	sseldd 3604	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐼) ∈ 𝐴)
26		onelon 5748	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝐼) ∈ 𝐴) → (𝐺‘𝐼) ∈ On)
27	2, 25, 26	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝐼) ∈ On)
28		oecl 7617	. . . . . . 7 ⊢ ((ω ∈ On ∧ (𝐺‘𝐼) ∈ On) → (ω ↑𝑜 (𝐺‘𝐼)) ∈ On)
29	1, 27, 28	sylancr 695	. . . . . 6 ⊢ (𝜑 → (ω ↑𝑜 (𝐺‘𝐼)) ∈ On)
30	16, 25	ffvelrnd 6360	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝐺‘𝐼)) ∈ ω)
31		nnon 7071	. . . . . . 7 ⊢ ((𝐹‘(𝐺‘𝐼)) ∈ ω → (𝐹‘(𝐺‘𝐼)) ∈ On)
32	30, 31	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐺‘𝐼)) ∈ On)
33		omcl 7616	. . . . . 6 ⊢ (((ω ↑𝑜 (𝐺‘𝐼)) ∈ On ∧ (𝐹‘(𝐺‘𝐼)) ∈ On) → ((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))) ∈ On)
34	29, 32, 33	syl2anc 693	. . . . 5 ⊢ (𝜑 → ((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))) ∈ On)
35	5, 6, 2, 21, 13	cantnfcl 8564	. . . . . . . 8 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
36	35	simprd 479	. . . . . . 7 ⊢ (𝜑 → dom 𝐺 ∈ ω)
37		elnn 7075	. . . . . . 7 ⊢ ((𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω) → 𝐼 ∈ ω)
38	20, 36, 37	syl2anc 693	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ω)
39		cnfcom.h	. . . . . . . 8 ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
40	39	cantnfvalf 8562	. . . . . . 7 ⊢ 𝐻:ω⟶On
41	40	ffvelrni 6358	. . . . . 6 ⊢ (𝐼 ∈ ω → (𝐻‘𝐼) ∈ On)
42	38, 41	syl 17	. . . . 5 ⊢ (𝜑 → (𝐻‘𝐼) ∈ On)
43		eqid 2622	. . . . . 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)))
44	43	oacomf1o 7645	. . . . 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)))))
45	34, 42, 44	syl2anc 693	. . . 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)))))
46		cnfcom.t	. . . . . . . 8 ⊢ 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
47	46	seqomsuc 7552	. . . . . . 7 ⊢ (𝐼 ∈ ω → (𝑇‘suc 𝐼) = (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇‘𝐼)))
48	38, 47	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑇‘suc 𝐼) = (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇‘𝐼)))
49		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑢𝐾
50		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑣𝐾
51		nfcv 2764	. . . . . . . . 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)))
52		nfcv 2764	. . . . . . . . 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)))
53		cnfcom.k	. . . . . . . . . 10 ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
54		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (dom 𝑓 +𝑜 𝑥) = (dom 𝑓 +𝑜 𝑦))
55	54	cbvmptv 4750	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) = (𝑦 ∈ 𝑀 ↦ (dom 𝑓 +𝑜 𝑦))
56		cnfcom.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = ((ω ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘)))
57		simpl 473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → 𝑘 = 𝑢)
58	57	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝐺‘𝑘) = (𝐺‘𝑢))
59	58	oveq2d 6666	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (ω ↑𝑜 (𝐺‘𝑘)) = (ω ↑𝑜 (𝐺‘𝑢)))
60	58	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝐹‘(𝐺‘𝑘)) = (𝐹‘(𝐺‘𝑢)))
61	59, 60	oveq12d 6668	. . . . . . . . . . . . . 14 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → ((ω ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) = ((ω ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))))
62	56, 61	syl5eq 2668	. . . . . . . . . . . . 13 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → 𝑀 = ((ω ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))))
63		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → 𝑓 = 𝑣)
64	63	dmeqd 5326	. . . . . . . . . . . . . 14 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → dom 𝑓 = dom 𝑣)
65	64	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (dom 𝑓 +𝑜 𝑦) = (dom 𝑣 +𝑜 𝑦))
66	62, 65	mpteq12dv 4733	. . . . . . . . . . . 12 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝑦 ∈ 𝑀 ↦ (dom 𝑓 +𝑜 𝑦)) = (𝑦 ∈ ((ω ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)))
67	55, 66	syl5eq 2668	. . . . . . . . . . 11 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝑥 ∈ 𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) = (𝑦 ∈ ((ω ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)))
68		oveq2 6658	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑀 +𝑜 𝑥) = (𝑀 +𝑜 𝑦))
69	68	cbvmptv 4750	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑦))
70	62	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝑀 +𝑜 𝑦) = (((ω ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) +𝑜 𝑦))
71	64, 70	mpteq12dv 4733	. . . . . . . . . . . . 13 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝑦 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑦)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) +𝑜 𝑦)))
72	69, 71	syl5eq 2668	. . . . . . . . . . . 12 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) +𝑜 𝑦)))
73	72	cnveqd 5298	. . . . . . . . . . 11 ⊢ ((𝑘 = 𝑢 ∧ 𝑓 = 𝑣) → ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = ◡(𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) +𝑜 𝑦)))
74	67, 73	uneq12d 3768	. . . . . . . . . 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))))
75	53, 74	syl5eq 2668	. . . . . . . . 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))))
76	49, 50, 51, 52, 75	cbvmpt2 6734	. . . . . . . 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))))
77	76	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)))))
78		simprl 794	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → 𝑢 = 𝐼)
79	78	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → (𝐺‘𝑢) = (𝐺‘𝐼))
80	79	oveq2d 6666	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → (ω ↑𝑜 (𝐺‘𝑢)) = (ω ↑𝑜 (𝐺‘𝐼)))
81	79	fveq2d 6195	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → (𝐹‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝐼)))
82	80, 81	oveq12d 6668	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → ((ω ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) = ((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))))
83		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼)) → 𝑣 = (𝑇‘𝐼))
84	83	dmeqd 5326	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼)) → dom 𝑣 = dom (𝑇‘𝐼))
85		cnfcom.3	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇‘𝐼):(𝐻‘𝐼)–1-1-onto→𝑂)
86		f1odm 6141	. . . . . . . . . . . 12 ⊢ ((𝑇‘𝐼):(𝐻‘𝐼)–1-1-onto→𝑂 → dom (𝑇‘𝐼) = (𝐻‘𝐼))
87	85, 86	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑇‘𝐼) = (𝐻‘𝐼))
88	84, 87	sylan9eqr 2678	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → dom 𝑣 = (𝐻‘𝐼))
89	88	oveq1d 6665	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 = 𝐼 ∧ 𝑣 = (𝑇‘𝐼))) → (dom 𝑣 +𝑜 𝑦) = ((𝐻‘𝐼) +𝑜 𝑦))
90	82, 89	mpteq12dv 4733	. . . . . . . 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)))
91	82	oveq1d 6665	. . . . . . . . . 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))
92	88, 91	mpteq12dv 4733	. . . . . . . . 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)))
93	92	cnveqd 5298	. . . . . . . 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)))
94	90, 93	uneq12d 3768	. . . . . . 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))))
95		elex 3212	. . . . . . . 8 ⊢ (𝐼 ∈ dom 𝐺 → 𝐼 ∈ V)
96	20, 95	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V)
97		fvexd 6203	. . . . . . 7 ⊢ (𝜑 → (𝑇‘𝐼) ∈ V)
98		ovex 6678	. . . . . . . . . 10 ⊢ ((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))) ∈ V
99	98	mptex 6486	. . . . . . . . 9 ⊢ (𝑦 ∈ ((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))) ↦ ((𝐻‘𝐼) +𝑜 𝑦)) ∈ V
100		fvex 6201	. . . . . . . . . . 11 ⊢ (𝐻‘𝐼) ∈ V
101	100	mptex 6486	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))) +𝑜 𝑦)) ∈ V
102	101	cnvex 7113	. . . . . . . . 9 ⊢ ◡(𝑦 ∈ (𝐻‘𝐼) ↦ (((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))) +𝑜 𝑦)) ∈ V
103	99, 102	unex 6956	. . . . . . . 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
104	103	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)
105	77, 94, 96, 97, 104	ovmpt2d 6788	. . . . . 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))))
106	48, 105	eqtrd 2656	. . . . 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))))
107		f1oeq1 6127	. . . . 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))))))
108	106, 107	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))))))
109	45, 108	mpbird 247	. . 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)))))
110	1	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) → ω ∈ On)
111		simpl 473	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) → 𝐴 ∈ On)
112		simpr 477	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) → 𝐹 ∈ 𝑆)
113	56	oveq1i 6660	. . . . . . . . . 10 ⊢ (𝑀 +𝑜 𝑧) = (((ω ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)
114	113	a1i 11	. . . . . . . . 9 ⊢ ((𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑀 +𝑜 𝑧) = (((ω ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))
115	114	mpt2eq3ia 6720	. . . . . . . 8 ⊢ (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))
116		eqid 2622	. . . . . . . 8 ⊢ ∅ = ∅
117		seqomeq12 7549	. . . . . . . 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)), ∅))
118	115, 116, 117	mp2an 708	. . . . . . 7 ⊢ seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)
119	39, 118	eqtri 2644	. . . . . 6 ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)
120	5, 110, 111, 21, 112, 119	cantnfsuc 8567	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ 𝐼 ∈ ω) → (𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))) +𝑜 (𝐻‘𝐼)))
121	2, 13, 38, 120	syl21anc 1325	. . . 4 ⊢ (𝜑 → (𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))) +𝑜 (𝐻‘𝐼)))
122		f1oeq2 6128	. . . 4 ⊢ ((𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))) +𝑜 (𝐻‘𝐼)) → ((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))))))
123	121, 122	syl 17	. . 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))))))
124	109, 123	mpbird 247	. 2 ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻‘𝐼) +𝑜 ((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼)))))
125		sssucid 5802	. . . . . 6 ⊢ dom 𝐺 ⊆ suc dom 𝐺
126	125, 20	sseldi 3601	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ suc dom 𝐺)
127		epelg 5030	. . . . . . . . . . 11 ⊢ (𝐼 ∈ dom 𝐺 → (𝑦 E 𝐼 ↔ 𝑦 ∈ 𝐼))
128	20, 127	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 E 𝐼 ↔ 𝑦 ∈ 𝐼))
129	128	biimpar 502	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝑦 E 𝐼)
130		ovexd 6680	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V)
131	35	simpld 475	. . . . . . . . . . . 12 ⊢ (𝜑 → E We (𝐹 supp ∅))
132	21	oiiso 8442	. . . . . . . . . . . 12 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
133	130, 131, 132	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
134	133	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
135	21	oicl 8434	. . . . . . . . . . . 12 ⊢ Ord dom 𝐺
136		ordelss 5739	. . . . . . . . . . . 12 ⊢ ((Ord dom 𝐺 ∧ 𝐼 ∈ dom 𝐺) → 𝐼 ⊆ dom 𝐺)
137	135, 20, 136	sylancr 695	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ dom 𝐺)
138	137	sselda 3603	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ dom 𝐺)
139	20	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ dom 𝐺)
140		isorel 6576	. . . . . . . . . 10 ⊢ ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) ∧ (𝑦 ∈ dom 𝐺 ∧ 𝐼 ∈ dom 𝐺)) → (𝑦 E 𝐼 ↔ (𝐺‘𝑦) E (𝐺‘𝐼)))
141	134, 138, 139, 140	syl12anc 1324	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑦 E 𝐼 ↔ (𝐺‘𝑦) E (𝐺‘𝐼)))
142	129, 141	mpbid 222	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐺‘𝑦) E (𝐺‘𝐼))
143		fvex 6201	. . . . . . . . 9 ⊢ (𝐺‘𝐼) ∈ V
144	143	epelc 5031	. . . . . . . 8 ⊢ ((𝐺‘𝑦) E (𝐺‘𝐼) ↔ (𝐺‘𝑦) ∈ (𝐺‘𝐼))
145	142, 144	sylib 208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐺‘𝑦) ∈ (𝐺‘𝐼))
146	145	ralrimiva 2966	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 (𝐺‘𝑦) ∈ (𝐺‘𝐼))
147		ffun 6048	. . . . . . . 8 ⊢ (𝐺:dom 𝐺⟶(𝐹 supp ∅) → Fun 𝐺)
148	22, 147	ax-mp 5	. . . . . . 7 ⊢ Fun 𝐺
149		funimass4 6247	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐼 ⊆ dom 𝐺) → ((𝐺 “ 𝐼) ⊆ (𝐺‘𝐼) ↔ ∀𝑦 ∈ 𝐼 (𝐺‘𝑦) ∈ (𝐺‘𝐼)))
150	148, 137, 149	sylancr 695	. . . . . 6 ⊢ (𝜑 → ((𝐺 “ 𝐼) ⊆ (𝐺‘𝐼) ↔ ∀𝑦 ∈ 𝐼 (𝐺‘𝑦) ∈ (𝐺‘𝐼)))
151	146, 150	mpbird 247	. . . . 5 ⊢ (𝜑 → (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))
152	1	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → ω ∈ On)
153		simpll 790	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → 𝐴 ∈ On)
154		simplr 792	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → 𝐹 ∈ 𝑆)
155		peano1 7085	. . . . . . 7 ⊢ ∅ ∈ ω
156	155	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → ∅ ∈ ω)
157		simpr1 1067	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → 𝐼 ∈ suc dom 𝐺)
158		simpr2 1068	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → (𝐺‘𝐼) ∈ On)
159		simpr3 1069	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))
160	5, 152, 153, 21, 154, 119, 156, 157, 158, 159	cantnflt 8569	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐹 ∈ 𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺‘𝐼) ∈ On ∧ (𝐺 “ 𝐼) ⊆ (𝐺‘𝐼))) → (𝐻‘𝐼) ∈ (ω ↑𝑜 (𝐺‘𝐼)))
161	2, 13, 126, 27, 151, 160	syl23anc 1333	. . . 4 ⊢ (𝜑 → (𝐻‘𝐼) ∈ (ω ↑𝑜 (𝐺‘𝐼)))
162		ffn 6045	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶ω → 𝐹 Fn 𝐴)
163	16, 162	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐴)
164		0ex 4790	. . . . . . . . . 10 ⊢ ∅ ∈ V
165	164	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ V)
166		elsuppfn 7303	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ On ∧ ∅ ∈ V) → ((𝐺‘𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺‘𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝐼)) ≠ ∅)))
167	163, 2, 165, 166	syl3anc 1326	. . . . . . . 8 ⊢ (𝜑 → ((𝐺‘𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺‘𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝐼)) ≠ ∅)))
168		simpr 477	. . . . . . . 8 ⊢ (((𝐺‘𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝐼)) ≠ ∅) → (𝐹‘(𝐺‘𝐼)) ≠ ∅)
169	167, 168	syl6bi 243	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝐼) ∈ (𝐹 supp ∅) → (𝐹‘(𝐺‘𝐼)) ≠ ∅))
170	24, 169	mpd 15	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐺‘𝐼)) ≠ ∅)
171		on0eln0 5780	. . . . . . 7 ⊢ ((𝐹‘(𝐺‘𝐼)) ∈ On → (∅ ∈ (𝐹‘(𝐺‘𝐼)) ↔ (𝐹‘(𝐺‘𝐼)) ≠ ∅))
172	32, 171	syl 17	. . . . . 6 ⊢ (𝜑 → (∅ ∈ (𝐹‘(𝐺‘𝐼)) ↔ (𝐹‘(𝐺‘𝐼)) ≠ ∅))
173	170, 172	mpbird 247	. . . . 5 ⊢ (𝜑 → ∅ ∈ (𝐹‘(𝐺‘𝐼)))
174		omword1 7653	. . . . 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))))
175	29, 32, 173, 174	syl21anc 1325	. . . 4 ⊢ (𝜑 → (ω ↑𝑜 (𝐺‘𝐼)) ⊆ ((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))))
176		oaabs2 7725	. . . 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))))
177	161, 34, 175, 176	syl21anc 1325	. . 3 ⊢ (𝜑 → ((𝐻‘𝐼) +𝑜 ((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼)))) = ((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))))
178		f1oeq3 6129	. . 3 ⊢ (((𝐻‘𝐼) +𝑜 ((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼)))) = ((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))) → ((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)))))
179	177, 178	syl 17	. 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)))))
180	124, 179	mpbid 222	1 ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺‘𝐼)) ·𝑜 (𝐹‘(𝐺‘𝐼))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653   ↦ cmpt 4729   E cep 5028   We wwe 5072  ^◡ccnv 5113  dom cdm 5114   “ cima 5117  Ord word 5722  Oncon0 5723  suc csuc 5725  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888   Isom wiso 5889  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065   supp csupp 7295  seq_𝜔cseqom 7542   +_𝑜 coa 7557   ·_𝑜 comu 7558   ↑_𝑜 coe 7559   finSupp cfsupp 8275  OrdIsocoi 8414   CNF ccnf 8558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-cnf 8559

This theorem is referenced by:  cnfcom  8597