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Theorem cnfcomlem 8760
Description: Lemma for cnfcom 8761. (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 (𝜑𝐴 ∈ On)
cnfcom.b (𝜑𝐵 ∈ (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 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))
cnfcom.k 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
cnfcom.1 (𝜑𝐼 ∈ dom 𝐺)
cnfcom.2 (𝜑𝑂 ∈ (ω ↑𝑜 (𝐺𝐼)))
cnfcom.3 (𝜑 → (𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂)
Assertion
Ref Expression
cnfcomlem (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
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.
StepHypRef Expression
1 omelon 8707 . . . . . . 7 ω ∈ On
2 cnfcom.a . . . . . . . 8 (𝜑𝐴 ∈ On)
3 suppssdm 7459 . . . . . . . . . 10 (𝐹 supp ∅) ⊆ dom 𝐹
4 cnfcom.f . . . . . . . . . . . . . 14 𝐹 = ((ω CNF 𝐴)‘𝐵)
5 cnfcom.s . . . . . . . . . . . . . . . . 17 𝑆 = dom (ω CNF 𝐴)
61a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ω ∈ On)
75, 6, 2cantnff1o 8757 . . . . . . . . . . . . . . . 16 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴))
8 f1ocnv 6290 . . . . . . . . . . . . . . . 16 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) → (ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆)
9 f1of 6278 . . . . . . . . . . . . . . . 16 ((ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
107, 8, 93syl 18 . . . . . . . . . . . . . . 15 (𝜑(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
11 cnfcom.b . . . . . . . . . . . . . . 15 (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))
1210, 11ffvelrnd 6503 . . . . . . . . . . . . . 14 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
134, 12syl5eqel 2854 . . . . . . . . . . . . 13 (𝜑𝐹𝑆)
145, 6, 2cantnfs 8727 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
1513, 14mpbid 222 . . . . . . . . . . . 12 (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
1615simpld 482 . . . . . . . . . . 11 (𝜑𝐹:𝐴⟶ω)
17 fdm 6191 . . . . . . . . . . 11 (𝐹:𝐴⟶ω → dom 𝐹 = 𝐴)
1816, 17syl 17 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
193, 18syl5sseq 3802 . . . . . . . . 9 (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
20 cnfcom.1 . . . . . . . . . 10 (𝜑𝐼 ∈ dom 𝐺)
21 cnfcom.g . . . . . . . . . . . 12 𝐺 = OrdIso( E , (𝐹 supp ∅))
2221oif 8591 . . . . . . . . . . 11 𝐺:dom 𝐺⟶(𝐹 supp ∅)
2322ffvelrni 6501 . . . . . . . . . 10 (𝐼 ∈ dom 𝐺 → (𝐺𝐼) ∈ (𝐹 supp ∅))
2420, 23syl 17 . . . . . . . . 9 (𝜑 → (𝐺𝐼) ∈ (𝐹 supp ∅))
2519, 24sseldd 3753 . . . . . . . 8 (𝜑 → (𝐺𝐼) ∈ 𝐴)
26 onelon 5891 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐺𝐼) ∈ 𝐴) → (𝐺𝐼) ∈ On)
272, 25, 26syl2anc 573 . . . . . . 7 (𝜑 → (𝐺𝐼) ∈ On)
28 oecl 7771 . . . . . . 7 ((ω ∈ On ∧ (𝐺𝐼) ∈ On) → (ω ↑𝑜 (𝐺𝐼)) ∈ On)
291, 27, 28sylancr 575 . . . . . 6 (𝜑 → (ω ↑𝑜 (𝐺𝐼)) ∈ On)
3016, 25ffvelrnd 6503 . . . . . . 7 (𝜑 → (𝐹‘(𝐺𝐼)) ∈ ω)
31 nnon 7218 . . . . . . 7 ((𝐹‘(𝐺𝐼)) ∈ ω → (𝐹‘(𝐺𝐼)) ∈ On)
3230, 31syl 17 . . . . . 6 (𝜑 → (𝐹‘(𝐺𝐼)) ∈ On)
33 omcl 7770 . . . . . 6 (((ω ↑𝑜 (𝐺𝐼)) ∈ On ∧ (𝐹‘(𝐺𝐼)) ∈ On) → ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ On)
3429, 32, 33syl2anc 573 . . . . 5 (𝜑 → ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ On)
355, 6, 2, 21, 13cantnfcl 8728 . . . . . . . 8 (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
3635simprd 483 . . . . . . 7 (𝜑 → dom 𝐺 ∈ ω)
37 elnn 7222 . . . . . . 7 ((𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω) → 𝐼 ∈ ω)
3820, 36, 37syl2anc 573 . . . . . 6 (𝜑𝐼 ∈ ω)
39 cnfcom.h . . . . . . . 8 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
4039cantnfvalf 8726 . . . . . . 7 𝐻:ω⟶On
4140ffvelrni 6501 . . . . . 6 (𝐼 ∈ ω → (𝐻𝐼) ∈ On)
4238, 41syl 17 . . . . 5 (𝜑 → (𝐻𝐼) ∈ On)
43 eqid 2771 . . . . . 6 ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))) = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)))
4443oacomf1o 7799 . . . . 5 ((((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ On ∧ (𝐻𝐼) ∈ On) → ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
4534, 42, 44syl2anc 573 . . . 4 (𝜑 → ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
46 cnfcom.t . . . . . . . 8 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
4746seqomsuc 7705 . . . . . . 7 (𝐼 ∈ ω → (𝑇‘suc 𝐼) = (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇𝐼)))
4838, 47syl 17 . . . . . 6 (𝜑 → (𝑇‘suc 𝐼) = (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇𝐼)))
49 nfcv 2913 . . . . . . . . 9 𝑢𝐾
50 nfcv 2913 . . . . . . . . 9 𝑣𝐾
51 nfcv 2913 . . . . . . . . 9 𝑘((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
52 nfcv 2913 . . . . . . . . 9 𝑓((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
53 cnfcom.k . . . . . . . . . 10 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
54 oveq2 6801 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (dom 𝑓 +𝑜 𝑥) = (dom 𝑓 +𝑜 𝑦))
5554cbvmptv 4884 . . . . . . . . . . . 12 (𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) = (𝑦𝑀 ↦ (dom 𝑓 +𝑜 𝑦))
56 cnfcom.m . . . . . . . . . . . . . 14 𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))
57 simpl 468 . . . . . . . . . . . . . . . . 17 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑘 = 𝑢)
5857fveq2d 6336 . . . . . . . . . . . . . . . 16 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝐺𝑘) = (𝐺𝑢))
5958oveq2d 6809 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → (ω ↑𝑜 (𝐺𝑘)) = (ω ↑𝑜 (𝐺𝑢)))
6058fveq2d 6336 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝐹‘(𝐺𝑘)) = (𝐹‘(𝐺𝑢)))
6159, 60oveq12d 6811 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) = ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))))
6256, 61syl5eq 2817 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑀 = ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))))
63 simpr 471 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑓 = 𝑣)
6463dmeqd 5464 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → dom 𝑓 = dom 𝑣)
6564oveq1d 6808 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → (dom 𝑓 +𝑜 𝑦) = (dom 𝑣 +𝑜 𝑦))
6662, 65mpteq12dv 4867 . . . . . . . . . . . 12 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑦𝑀 ↦ (dom 𝑓 +𝑜 𝑦)) = (𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)))
6755, 66syl5eq 2817 . . . . . . . . . . 11 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) = (𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)))
68 oveq2 6801 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑀 +𝑜 𝑥) = (𝑀 +𝑜 𝑦))
6968cbvmptv 4884 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑦))
7062oveq1d 6808 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑀 +𝑜 𝑦) = (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦))
7164, 70mpteq12dv 4867 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑦 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑦)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
7269, 71syl5eq 2817 . . . . . . . . . . . 12 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
7372cnveqd 5436 . . . . . . . . . . 11 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
7467, 73uneq12d 3919 . . . . . . . . . 10 ((𝑘 = 𝑢𝑓 = 𝑣) → ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥))) = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦))))
7553, 74syl5eq 2817 . . . . . . . . 9 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝐾 = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦))))
7649, 50, 51, 52, 75cbvmpt2 6881 . . . . . . . 8 (𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾) = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦))))
7776a1i 11 . . . . . . 7 (𝜑 → (𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾) = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))))
78 simprl 754 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → 𝑢 = 𝐼)
7978fveq2d 6336 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝐺𝑢) = (𝐺𝐼))
8079oveq2d 6809 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (ω ↑𝑜 (𝐺𝑢)) = (ω ↑𝑜 (𝐺𝐼)))
8179fveq2d 6336 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝐹‘(𝐺𝑢)) = (𝐹‘(𝐺𝐼)))
8280, 81oveq12d 6811 . . . . . . . . 9 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
83 simpr 471 . . . . . . . . . . . 12 ((𝑢 = 𝐼𝑣 = (𝑇𝐼)) → 𝑣 = (𝑇𝐼))
8483dmeqd 5464 . . . . . . . . . . 11 ((𝑢 = 𝐼𝑣 = (𝑇𝐼)) → dom 𝑣 = dom (𝑇𝐼))
85 cnfcom.3 . . . . . . . . . . . 12 (𝜑 → (𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂)
86 f1odm 6282 . . . . . . . . . . . 12 ((𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂 → dom (𝑇𝐼) = (𝐻𝐼))
8785, 86syl 17 . . . . . . . . . . 11 (𝜑 → dom (𝑇𝐼) = (𝐻𝐼))
8884, 87sylan9eqr 2827 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → dom 𝑣 = (𝐻𝐼))
8988oveq1d 6808 . . . . . . . . 9 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (dom 𝑣 +𝑜 𝑦) = ((𝐻𝐼) +𝑜 𝑦))
9082, 89mpteq12dv 4867 . . . . . . . 8 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) = (𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)))
9182oveq1d 6808 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))
9288, 91mpteq12dv 4867 . . . . . . . . 9 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)) = (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)))
9392cnveqd 5436 . . . . . . . 8 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)) = (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)))
9490, 93uneq12d 3919 . . . . . . 7 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦))) = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))))
95 elex 3364 . . . . . . . 8 (𝐼 ∈ dom 𝐺𝐼 ∈ V)
9620, 95syl 17 . . . . . . 7 (𝜑𝐼 ∈ V)
97 fvexd 6344 . . . . . . 7 (𝜑 → (𝑇𝐼) ∈ V)
98 ovex 6823 . . . . . . . . . 10 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ V
9998mptex 6630 . . . . . . . . 9 (𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∈ V
100 fvex 6342 . . . . . . . . . . 11 (𝐻𝐼) ∈ V
101100mptex 6630 . . . . . . . . . 10 (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)) ∈ V
102101cnvex 7260 . . . . . . . . 9 (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)) ∈ V
10399, 102unex 7103 . . . . . . . 8 ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))) ∈ V
104103a1i 11 . . . . . . 7 (𝜑 → ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))) ∈ V)
10577, 94, 96, 97, 104ovmpt2d 6935 . . . . . 6 (𝜑 → (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇𝐼)) = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))))
10648, 105eqtrd 2805 . . . . 5 (𝜑 → (𝑇‘suc 𝐼) = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))))
107 f1oeq1 6268 . . . . 5 ((𝑇‘suc 𝐼) = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))) → ((𝑇‘suc 𝐼):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) ↔ ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))))
108106, 107syl 17 . . . 4 (𝜑 → ((𝑇‘suc 𝐼):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) ↔ ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))))
10945, 108mpbird 247 . . 3 (𝜑 → (𝑇‘suc 𝐼):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
1101a1i 11 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → ω ∈ On)
111 simpl 468 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → 𝐴 ∈ On)
112 simpr 471 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → 𝐹𝑆)
11356oveq1i 6803 . . . . . . . . . 10 (𝑀 +𝑜 𝑧) = (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)
114113a1i 11 . . . . . . . . 9 ((𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑀 +𝑜 𝑧) = (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
115114mpt2eq3ia 6867 . . . . . . . 8 (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
116 eqid 2771 . . . . . . . 8 ∅ = ∅
117 seqomeq12 7702 . . . . . . . 8 (((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)) ∧ ∅ = ∅) → seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅))
118115, 116, 117mp2an 672 . . . . . . 7 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
11939, 118eqtri 2793 . . . . . 6 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
1205, 110, 111, 21, 112, 119cantnfsuc 8731 . . . . 5 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ 𝐼 ∈ ω) → (𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼)))
1212, 13, 38, 120syl21anc 1475 . . . 4 (𝜑 → (𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼)))
122 f1oeq2 6269 . . . 4 ((𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼)) → ((𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) ↔ (𝑇‘suc 𝐼):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))))
123121, 122syl 17 . . 3 (𝜑 → ((𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) ↔ (𝑇‘suc 𝐼):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))))
124109, 123mpbird 247 . 2 (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
125 sssucid 5945 . . . . . 6 dom 𝐺 ⊆ suc dom 𝐺
126125, 20sseldi 3750 . . . . 5 (𝜑𝐼 ∈ suc dom 𝐺)
127 epelg 5163 . . . . . . . . . . 11 (𝐼 ∈ dom 𝐺 → (𝑦 E 𝐼𝑦𝐼))
12820, 127syl 17 . . . . . . . . . 10 (𝜑 → (𝑦 E 𝐼𝑦𝐼))
129128biimpar 463 . . . . . . . . 9 ((𝜑𝑦𝐼) → 𝑦 E 𝐼)
130 ovexd 6825 . . . . . . . . . . . 12 (𝜑 → (𝐹 supp ∅) ∈ V)
13135simpld 482 . . . . . . . . . . . 12 (𝜑 → E We (𝐹 supp ∅))
13221oiiso 8598 . . . . . . . . . . . 12 (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
133130, 131, 132syl2anc 573 . . . . . . . . . . 11 (𝜑𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
134133adantr 466 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
13521oicl 8590 . . . . . . . . . . . 12 Ord dom 𝐺
136 ordelss 5882 . . . . . . . . . . . 12 ((Ord dom 𝐺𝐼 ∈ dom 𝐺) → 𝐼 ⊆ dom 𝐺)
137135, 20, 136sylancr 575 . . . . . . . . . . 11 (𝜑𝐼 ⊆ dom 𝐺)
138137sselda 3752 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝑦 ∈ dom 𝐺)
13920adantr 466 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐼 ∈ dom 𝐺)
140 isorel 6719 . . . . . . . . . 10 ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) ∧ (𝑦 ∈ dom 𝐺𝐼 ∈ dom 𝐺)) → (𝑦 E 𝐼 ↔ (𝐺𝑦) E (𝐺𝐼)))
141134, 138, 139, 140syl12anc 1474 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑦 E 𝐼 ↔ (𝐺𝑦) E (𝐺𝐼)))
142129, 141mpbid 222 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝐺𝑦) E (𝐺𝐼))
143 fvex 6342 . . . . . . . . 9 (𝐺𝐼) ∈ V
144143epelc 5164 . . . . . . . 8 ((𝐺𝑦) E (𝐺𝐼) ↔ (𝐺𝑦) ∈ (𝐺𝐼))
145142, 144sylib 208 . . . . . . 7 ((𝜑𝑦𝐼) → (𝐺𝑦) ∈ (𝐺𝐼))
146145ralrimiva 3115 . . . . . 6 (𝜑 → ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼))
147 ffun 6188 . . . . . . . 8 (𝐺:dom 𝐺⟶(𝐹 supp ∅) → Fun 𝐺)
14822, 147ax-mp 5 . . . . . . 7 Fun 𝐺
149 funimass4 6389 . . . . . . 7 ((Fun 𝐺𝐼 ⊆ dom 𝐺) → ((𝐺𝐼) ⊆ (𝐺𝐼) ↔ ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼)))
150148, 137, 149sylancr 575 . . . . . 6 (𝜑 → ((𝐺𝐼) ⊆ (𝐺𝐼) ↔ ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼)))
151146, 150mpbird 247 . . . . 5 (𝜑 → (𝐺𝐼) ⊆ (𝐺𝐼))
1521a1i 11 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → ω ∈ On)
153 simpll 750 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐴 ∈ On)
154 simplr 752 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐹𝑆)
155 peano1 7232 . . . . . . 7 ∅ ∈ ω
156155a1i 11 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → ∅ ∈ ω)
157 simpr1 1233 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐼 ∈ suc dom 𝐺)
158 simpr2 1235 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐺𝐼) ∈ On)
159 simpr3 1237 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐺𝐼) ⊆ (𝐺𝐼))
1605, 152, 153, 21, 154, 119, 156, 157, 158, 159cantnflt 8733 . . . . 5 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐻𝐼) ∈ (ω ↑𝑜 (𝐺𝐼)))
1612, 13, 126, 27, 151, 160syl23anc 1483 . . . 4 (𝜑 → (𝐻𝐼) ∈ (ω ↑𝑜 (𝐺𝐼)))
162 ffn 6185 . . . . . . . . . 10 (𝐹:𝐴⟶ω → 𝐹 Fn 𝐴)
16316, 162syl 17 . . . . . . . . 9 (𝜑𝐹 Fn 𝐴)
164 0ex 4924 . . . . . . . . . 10 ∅ ∈ V
165164a1i 11 . . . . . . . . 9 (𝜑 → ∅ ∈ V)
166 elsuppfn 7454 . . . . . . . . 9 ((𝐹 Fn 𝐴𝐴 ∈ On ∧ ∅ ∈ V) → ((𝐺𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅)))
167163, 2, 165, 166syl3anc 1476 . . . . . . . 8 (𝜑 → ((𝐺𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅)))
168 simpr 471 . . . . . . . 8 (((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅) → (𝐹‘(𝐺𝐼)) ≠ ∅)
169167, 168syl6bi 243 . . . . . . 7 (𝜑 → ((𝐺𝐼) ∈ (𝐹 supp ∅) → (𝐹‘(𝐺𝐼)) ≠ ∅))
17024, 169mpd 15 . . . . . 6 (𝜑 → (𝐹‘(𝐺𝐼)) ≠ ∅)
171 on0eln0 5923 . . . . . . 7 ((𝐹‘(𝐺𝐼)) ∈ On → (∅ ∈ (𝐹‘(𝐺𝐼)) ↔ (𝐹‘(𝐺𝐼)) ≠ ∅))
17232, 171syl 17 . . . . . 6 (𝜑 → (∅ ∈ (𝐹‘(𝐺𝐼)) ↔ (𝐹‘(𝐺𝐼)) ≠ ∅))
173170, 172mpbird 247 . . . . 5 (𝜑 → ∅ ∈ (𝐹‘(𝐺𝐼)))
174 omword1 7807 . . . . 5 ((((ω ↑𝑜 (𝐺𝐼)) ∈ On ∧ (𝐹‘(𝐺𝐼)) ∈ On) ∧ ∅ ∈ (𝐹‘(𝐺𝐼))) → (ω ↑𝑜 (𝐺𝐼)) ⊆ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
17529, 32, 173, 174syl21anc 1475 . . . 4 (𝜑 → (ω ↑𝑜 (𝐺𝐼)) ⊆ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
176 oaabs2 7879 . . . 4 ((((𝐻𝐼) ∈ (ω ↑𝑜 (𝐺𝐼)) ∧ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ On) ∧ (ω ↑𝑜 (𝐺𝐼)) ⊆ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) → ((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
177161, 34, 175, 176syl21anc 1475 . . 3 (𝜑 → ((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
178 f1oeq3 6270 . . 3 (((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) → ((𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) ↔ (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
179177, 178syl 17 . 2 (𝜑 → ((𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) ↔ (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
180124, 179mpbid 222 1 (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  Vcvv 3351  cun 3721  wss 3723  c0 4063   class class class wbr 4786  cmpt 4863   E cep 5161   We wwe 5207  ccnv 5248  dom cdm 5249  cima 5252  Ord word 5865  Oncon0 5866  suc csuc 5868  Fun wfun 6025   Fn wfn 6026  wf 6027  1-1-ontowf1o 6030  cfv 6031   Isom wiso 6032  (class class class)co 6793  cmpt2 6795  ωcom 7212   supp csupp 7446  seq𝜔cseqom 7695   +𝑜 coa 7710   ·𝑜 comu 7711  𝑜 coe 7712   finSupp cfsupp 8431  OrdIsocoi 8570   CNF ccnf 8722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seqom 7696  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718  df-oexp 7719  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-oi 8571  df-cnf 8723
This theorem is referenced by:  cnfcom  8761
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