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Theorem cnfcomlem 8540
Description: Lemma for cnfcom 8541. (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 8487 . . . . . . 7 ω ∈ On
2 cnfcom.a . . . . . . . 8 (𝜑𝐴 ∈ On)
3 suppssdm 7253 . . . . . . . . . 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 8537 . . . . . . . . . . . . . . . 16 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴))
8 f1ocnv 6106 . . . . . . . . . . . . . . . 16 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) → (ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆)
9 f1of 6094 . . . . . . . . . . . . . . . 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 6316 . . . . . . . . . . . . . 14 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
134, 12syl5eqel 2702 . . . . . . . . . . . . 13 (𝜑𝐹𝑆)
145, 6, 2cantnfs 8507 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
1513, 14mpbid 222 . . . . . . . . . . . 12 (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
1615simpld 475 . . . . . . . . . . 11 (𝜑𝐹:𝐴⟶ω)
17 fdm 6008 . . . . . . . . . . 11 (𝐹:𝐴⟶ω → dom 𝐹 = 𝐴)
1816, 17syl 17 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
193, 18syl5sseq 3632 . . . . . . . . 9 (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
20 cnfcom.1 . . . . . . . . . 10 (𝜑𝐼 ∈ dom 𝐺)
21 cnfcom.g . . . . . . . . . . . 12 𝐺 = OrdIso( E , (𝐹 supp ∅))
2221oif 8379 . . . . . . . . . . 11 𝐺:dom 𝐺⟶(𝐹 supp ∅)
2322ffvelrni 6314 . . . . . . . . . 10 (𝐼 ∈ dom 𝐺 → (𝐺𝐼) ∈ (𝐹 supp ∅))
2420, 23syl 17 . . . . . . . . 9 (𝜑 → (𝐺𝐼) ∈ (𝐹 supp ∅))
2519, 24sseldd 3584 . . . . . . . 8 (𝜑 → (𝐺𝐼) ∈ 𝐴)
26 onelon 5707 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐺𝐼) ∈ 𝐴) → (𝐺𝐼) ∈ On)
272, 25, 26syl2anc 692 . . . . . . 7 (𝜑 → (𝐺𝐼) ∈ On)
28 oecl 7562 . . . . . . 7 ((ω ∈ On ∧ (𝐺𝐼) ∈ On) → (ω ↑𝑜 (𝐺𝐼)) ∈ On)
291, 27, 28sylancr 694 . . . . . 6 (𝜑 → (ω ↑𝑜 (𝐺𝐼)) ∈ On)
3016, 25ffvelrnd 6316 . . . . . . 7 (𝜑 → (𝐹‘(𝐺𝐼)) ∈ ω)
31 nnon 7018 . . . . . . 7 ((𝐹‘(𝐺𝐼)) ∈ ω → (𝐹‘(𝐺𝐼)) ∈ On)
3230, 31syl 17 . . . . . 6 (𝜑 → (𝐹‘(𝐺𝐼)) ∈ On)
33 omcl 7561 . . . . . 6 (((ω ↑𝑜 (𝐺𝐼)) ∈ On ∧ (𝐹‘(𝐺𝐼)) ∈ On) → ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ On)
3429, 32, 33syl2anc 692 . . . . 5 (𝜑 → ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ On)
355, 6, 2, 21, 13cantnfcl 8508 . . . . . . . 8 (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
3635simprd 479 . . . . . . 7 (𝜑 → dom 𝐺 ∈ ω)
37 elnn 7022 . . . . . . 7 ((𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω) → 𝐼 ∈ ω)
3820, 36, 37syl2anc 692 . . . . . 6 (𝜑𝐼 ∈ ω)
39 cnfcom.h . . . . . . . 8 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
4039cantnfvalf 8506 . . . . . . 7 𝐻:ω⟶On
4140ffvelrni 6314 . . . . . 6 (𝐼 ∈ ω → (𝐻𝐼) ∈ On)
4238, 41syl 17 . . . . 5 (𝜑 → (𝐻𝐼) ∈ On)
43 eqid 2621 . . . . . 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 7590 . . . . 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 692 . . . 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 7497 . . . . . . 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 2761 . . . . . . . . 9 𝑢𝐾
50 nfcv 2761 . . . . . . . . 9 𝑣𝐾
51 nfcv 2761 . . . . . . . . 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 2761 . . . . . . . . 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 6612 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (dom 𝑓 +𝑜 𝑥) = (dom 𝑓 +𝑜 𝑦))
5554cbvmptv 4710 . . . . . . . . . . . 12 (𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) = (𝑦𝑀 ↦ (dom 𝑓 +𝑜 𝑦))
56 cnfcom.m . . . . . . . . . . . . . 14 𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))
57 simpl 473 . . . . . . . . . . . . . . . . 17 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑘 = 𝑢)
5857fveq2d 6152 . . . . . . . . . . . . . . . 16 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝐺𝑘) = (𝐺𝑢))
5958oveq2d 6620 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → (ω ↑𝑜 (𝐺𝑘)) = (ω ↑𝑜 (𝐺𝑢)))
6058fveq2d 6152 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝐹‘(𝐺𝑘)) = (𝐹‘(𝐺𝑢)))
6159, 60oveq12d 6622 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) = ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))))
6256, 61syl5eq 2667 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑀 = ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))))
63 simpr 477 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑓 = 𝑣)
6463dmeqd 5286 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → dom 𝑓 = dom 𝑣)
6564oveq1d 6619 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → (dom 𝑓 +𝑜 𝑦) = (dom 𝑣 +𝑜 𝑦))
6662, 65mpteq12dv 4693 . . . . . . . . . . . 12 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑦𝑀 ↦ (dom 𝑓 +𝑜 𝑦)) = (𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)))
6755, 66syl5eq 2667 . . . . . . . . . . 11 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) = (𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)))
68 oveq2 6612 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑀 +𝑜 𝑥) = (𝑀 +𝑜 𝑦))
6968cbvmptv 4710 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑦))
7062oveq1d 6619 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑀 +𝑜 𝑦) = (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦))
7164, 70mpteq12dv 4693 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑦 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑦)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
7269, 71syl5eq 2667 . . . . . . . . . . . 12 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
7372cnveqd 5258 . . . . . . . . . . 11 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
7467, 73uneq12d 3746 . . . . . . . . . 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 2667 . . . . . . . . 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 6687 . . . . . . . 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 793 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → 𝑢 = 𝐼)
7978fveq2d 6152 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝐺𝑢) = (𝐺𝐼))
8079oveq2d 6620 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (ω ↑𝑜 (𝐺𝑢)) = (ω ↑𝑜 (𝐺𝐼)))
8179fveq2d 6152 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝐹‘(𝐺𝑢)) = (𝐹‘(𝐺𝐼)))
8280, 81oveq12d 6622 . . . . . . . . 9 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
83 simpr 477 . . . . . . . . . . . 12 ((𝑢 = 𝐼𝑣 = (𝑇𝐼)) → 𝑣 = (𝑇𝐼))
8483dmeqd 5286 . . . . . . . . . . 11 ((𝑢 = 𝐼𝑣 = (𝑇𝐼)) → dom 𝑣 = dom (𝑇𝐼))
85 cnfcom.3 . . . . . . . . . . . 12 (𝜑 → (𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂)
86 f1odm 6098 . . . . . . . . . . . 12 ((𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂 → dom (𝑇𝐼) = (𝐻𝐼))
8785, 86syl 17 . . . . . . . . . . 11 (𝜑 → dom (𝑇𝐼) = (𝐻𝐼))
8884, 87sylan9eqr 2677 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → dom 𝑣 = (𝐻𝐼))
8988oveq1d 6619 . . . . . . . . 9 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (dom 𝑣 +𝑜 𝑦) = ((𝐻𝐼) +𝑜 𝑦))
9082, 89mpteq12dv 4693 . . . . . . . 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 6619 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))
9288, 91mpteq12dv 4693 . . . . . . . . 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 5258 . . . . . . . 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 3746 . . . . . . 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 3198 . . . . . . . 8 (𝐼 ∈ dom 𝐺𝐼 ∈ V)
9620, 95syl 17 . . . . . . 7 (𝜑𝐼 ∈ V)
97 fvex 6158 . . . . . . . 8 (𝑇𝐼) ∈ V
9897a1i 11 . . . . . . 7 (𝜑 → (𝑇𝐼) ∈ V)
99 ovex 6632 . . . . . . . . . 10 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ V
10099mptex 6440 . . . . . . . . 9 (𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∈ V
101 fvex 6158 . . . . . . . . . . 11 (𝐻𝐼) ∈ V
102101mptex 6440 . . . . . . . . . 10 (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)) ∈ V
103102cnvex 7060 . . . . . . . . 9 (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)) ∈ V
104100, 103unex 6909 . . . . . . . 8 ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))) ∈ V
105104a1i 11 . . . . . . 7 (𝜑 → ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))) ∈ V)
10677, 94, 96, 98, 105ovmpt2d 6741 . . . . . 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))))
10748, 106eqtrd 2655 . . . . 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))))
108 f1oeq1 6084 . . . . 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))))))
109107, 108syl 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))))))
11045, 109mpbird 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)))))
1111a1i 11 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → ω ∈ On)
112 simpl 473 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → 𝐴 ∈ On)
113 simpr 477 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → 𝐹𝑆)
11456oveq1i 6614 . . . . . . . . . 10 (𝑀 +𝑜 𝑧) = (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)
115114a1i 11 . . . . . . . . 9 ((𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑀 +𝑜 𝑧) = (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
116115mpt2eq3ia 6673 . . . . . . . 8 (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
117 eqid 2621 . . . . . . . 8 ∅ = ∅
118 seqomeq12 7494 . . . . . . . 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)), ∅))
119116, 117, 118mp2an 707 . . . . . . 7 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
12039, 119eqtri 2643 . . . . . 6 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
1215, 111, 112, 21, 113, 120cantnfsuc 8511 . . . . 5 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ 𝐼 ∈ ω) → (𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼)))
1222, 13, 38, 121syl21anc 1322 . . . 4 (𝜑 → (𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼)))
123 f1oeq2 6085 . . . 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))))))
124122, 123syl 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))))))
125110, 124mpbird 247 . 2 (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
126 sssucid 5761 . . . . . 6 dom 𝐺 ⊆ suc dom 𝐺
127126, 20sseldi 3581 . . . . 5 (𝜑𝐼 ∈ suc dom 𝐺)
128 epelg 4986 . . . . . . . . . . 11 (𝐼 ∈ dom 𝐺 → (𝑦 E 𝐼𝑦𝐼))
12920, 128syl 17 . . . . . . . . . 10 (𝜑 → (𝑦 E 𝐼𝑦𝐼))
130129biimpar 502 . . . . . . . . 9 ((𝜑𝑦𝐼) → 𝑦 E 𝐼)
1312, 19ssexd 4765 . . . . . . . . . . . 12 (𝜑 → (𝐹 supp ∅) ∈ V)
13235simpld 475 . . . . . . . . . . . 12 (𝜑 → E We (𝐹 supp ∅))
13321oiiso 8386 . . . . . . . . . . . 12 (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
134131, 132, 133syl2anc 692 . . . . . . . . . . 11 (𝜑𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
135134adantr 481 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
13621oicl 8378 . . . . . . . . . . . 12 Ord dom 𝐺
137 ordelss 5698 . . . . . . . . . . . 12 ((Ord dom 𝐺𝐼 ∈ dom 𝐺) → 𝐼 ⊆ dom 𝐺)
138136, 20, 137sylancr 694 . . . . . . . . . . 11 (𝜑𝐼 ⊆ dom 𝐺)
139138sselda 3583 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝑦 ∈ dom 𝐺)
14020adantr 481 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐼 ∈ dom 𝐺)
141 isorel 6530 . . . . . . . . . 10 ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) ∧ (𝑦 ∈ dom 𝐺𝐼 ∈ dom 𝐺)) → (𝑦 E 𝐼 ↔ (𝐺𝑦) E (𝐺𝐼)))
142135, 139, 140, 141syl12anc 1321 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑦 E 𝐼 ↔ (𝐺𝑦) E (𝐺𝐼)))
143130, 142mpbid 222 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝐺𝑦) E (𝐺𝐼))
144 fvex 6158 . . . . . . . . 9 (𝐺𝐼) ∈ V
145144epelc 4987 . . . . . . . 8 ((𝐺𝑦) E (𝐺𝐼) ↔ (𝐺𝑦) ∈ (𝐺𝐼))
146143, 145sylib 208 . . . . . . 7 ((𝜑𝑦𝐼) → (𝐺𝑦) ∈ (𝐺𝐼))
147146ralrimiva 2960 . . . . . 6 (𝜑 → ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼))
148 ffun 6005 . . . . . . . 8 (𝐺:dom 𝐺⟶(𝐹 supp ∅) → Fun 𝐺)
14922, 148ax-mp 5 . . . . . . 7 Fun 𝐺
150 funimass4 6204 . . . . . . 7 ((Fun 𝐺𝐼 ⊆ dom 𝐺) → ((𝐺𝐼) ⊆ (𝐺𝐼) ↔ ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼)))
151149, 138, 150sylancr 694 . . . . . 6 (𝜑 → ((𝐺𝐼) ⊆ (𝐺𝐼) ↔ ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼)))
152147, 151mpbird 247 . . . . 5 (𝜑 → (𝐺𝐼) ⊆ (𝐺𝐼))
1531a1i 11 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → ω ∈ On)
154 simpll 789 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐴 ∈ On)
155 simplr 791 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐹𝑆)
156 peano1 7032 . . . . . . 7 ∅ ∈ ω
157156a1i 11 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → ∅ ∈ ω)
158 simpr1 1065 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐼 ∈ suc dom 𝐺)
159 simpr2 1066 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐺𝐼) ∈ On)
160 simpr3 1067 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐺𝐼) ⊆ (𝐺𝐼))
1615, 153, 154, 21, 155, 120, 157, 158, 159, 160cantnflt 8513 . . . . 5 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐻𝐼) ∈ (ω ↑𝑜 (𝐺𝐼)))
1622, 13, 127, 27, 152, 161syl23anc 1330 . . . 4 (𝜑 → (𝐻𝐼) ∈ (ω ↑𝑜 (𝐺𝐼)))
163 ffn 6002 . . . . . . . . . 10 (𝐹:𝐴⟶ω → 𝐹 Fn 𝐴)
16416, 163syl 17 . . . . . . . . 9 (𝜑𝐹 Fn 𝐴)
165 0ex 4750 . . . . . . . . . 10 ∅ ∈ V
166165a1i 11 . . . . . . . . 9 (𝜑 → ∅ ∈ V)
167 elsuppfn 7248 . . . . . . . . 9 ((𝐹 Fn 𝐴𝐴 ∈ On ∧ ∅ ∈ V) → ((𝐺𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅)))
168164, 2, 166, 167syl3anc 1323 . . . . . . . 8 (𝜑 → ((𝐺𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅)))
169 simpr 477 . . . . . . . 8 (((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅) → (𝐹‘(𝐺𝐼)) ≠ ∅)
170168, 169syl6bi 243 . . . . . . 7 (𝜑 → ((𝐺𝐼) ∈ (𝐹 supp ∅) → (𝐹‘(𝐺𝐼)) ≠ ∅))
17124, 170mpd 15 . . . . . 6 (𝜑 → (𝐹‘(𝐺𝐼)) ≠ ∅)
172 on0eln0 5739 . . . . . . 7 ((𝐹‘(𝐺𝐼)) ∈ On → (∅ ∈ (𝐹‘(𝐺𝐼)) ↔ (𝐹‘(𝐺𝐼)) ≠ ∅))
17332, 172syl 17 . . . . . 6 (𝜑 → (∅ ∈ (𝐹‘(𝐺𝐼)) ↔ (𝐹‘(𝐺𝐼)) ≠ ∅))
174171, 173mpbird 247 . . . . 5 (𝜑 → ∅ ∈ (𝐹‘(𝐺𝐼)))
175 omword1 7598 . . . . 5 ((((ω ↑𝑜 (𝐺𝐼)) ∈ On ∧ (𝐹‘(𝐺𝐼)) ∈ On) ∧ ∅ ∈ (𝐹‘(𝐺𝐼))) → (ω ↑𝑜 (𝐺𝐼)) ⊆ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
17629, 32, 174, 175syl21anc 1322 . . . 4 (𝜑 → (ω ↑𝑜 (𝐺𝐼)) ⊆ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
177 oaabs2 7670 . . . 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))))
178162, 34, 176, 177syl21anc 1322 . . 3 (𝜑 → ((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
179 f1oeq3 6086 . . 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)))))
180178, 179syl 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)))))
181125, 180mpbid 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3186  cun 3553  wss 3555  c0 3891   class class class wbr 4613  cmpt 4673   E cep 4983   We wwe 5032  ccnv 5073  dom cdm 5074  cima 5077  Ord word 5681  Oncon0 5682  suc csuc 5684  Fun wfun 5841   Fn wfn 5842  wf 5843  1-1-ontowf1o 5846  cfv 5847   Isom wiso 5848  (class class class)co 6604  cmpt2 6606  ωcom 7012   supp csupp 7240  seq𝜔cseqom 7487   +𝑜 coa 7502   ·𝑜 comu 7503  𝑜 coe 7504   finSupp cfsupp 8219  OrdIsocoi 8358   CNF ccnf 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-oi 8359  df-cnf 8503
This theorem is referenced by:  cnfcom  8541
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