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Theorem cnfcomlem 8581
 Description: Lemma for cnfcom 8582. (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 8528 . . . . . . 7 ω ∈ On
2 cnfcom.a . . . . . . . 8 (𝜑𝐴 ∈ On)
3 suppssdm 7293 . . . . . . . . . 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 8578 . . . . . . . . . . . . . . . 16 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴))
8 f1ocnv 6136 . . . . . . . . . . . . . . . 16 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) → (ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆)
9 f1of 6124 . . . . . . . . . . . . . . . 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 6346 . . . . . . . . . . . . . 14 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
134, 12syl5eqel 2703 . . . . . . . . . . . . 13 (𝜑𝐹𝑆)
145, 6, 2cantnfs 8548 . . . . . . . . . . . . 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 6038 . . . . . . . . . . 11 (𝐹:𝐴⟶ω → dom 𝐹 = 𝐴)
1816, 17syl 17 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
193, 18syl5sseq 3645 . . . . . . . . 9 (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
20 cnfcom.1 . . . . . . . . . 10 (𝜑𝐼 ∈ dom 𝐺)
21 cnfcom.g . . . . . . . . . . . 12 𝐺 = OrdIso( E , (𝐹 supp ∅))
2221oif 8420 . . . . . . . . . . 11 𝐺:dom 𝐺⟶(𝐹 supp ∅)
2322ffvelrni 6344 . . . . . . . . . 10 (𝐼 ∈ dom 𝐺 → (𝐺𝐼) ∈ (𝐹 supp ∅))
2420, 23syl 17 . . . . . . . . 9 (𝜑 → (𝐺𝐼) ∈ (𝐹 supp ∅))
2519, 24sseldd 3596 . . . . . . . 8 (𝜑 → (𝐺𝐼) ∈ 𝐴)
26 onelon 5736 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐺𝐼) ∈ 𝐴) → (𝐺𝐼) ∈ On)
272, 25, 26syl2anc 692 . . . . . . 7 (𝜑 → (𝐺𝐼) ∈ On)
28 oecl 7602 . . . . . . 7 ((ω ∈ On ∧ (𝐺𝐼) ∈ On) → (ω ↑𝑜 (𝐺𝐼)) ∈ On)
291, 27, 28sylancr 694 . . . . . 6 (𝜑 → (ω ↑𝑜 (𝐺𝐼)) ∈ On)
3016, 25ffvelrnd 6346 . . . . . . 7 (𝜑 → (𝐹‘(𝐺𝐼)) ∈ ω)
31 nnon 7056 . . . . . . 7 ((𝐹‘(𝐺𝐼)) ∈ ω → (𝐹‘(𝐺𝐼)) ∈ On)
3230, 31syl 17 . . . . . 6 (𝜑 → (𝐹‘(𝐺𝐼)) ∈ On)
33 omcl 7601 . . . . . 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 8549 . . . . . . . 8 (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
3635simprd 479 . . . . . . 7 (𝜑 → dom 𝐺 ∈ ω)
37 elnn 7060 . . . . . . 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 8547 . . . . . . 7 𝐻:ω⟶On
4140ffvelrni 6344 . . . . . 6 (𝐼 ∈ ω → (𝐻𝐼) ∈ On)
4238, 41syl 17 . . . . 5 (𝜑 → (𝐻𝐼) ∈ On)
43 eqid 2620 . . . . . 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 7630 . . . . 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 7537 . . . . . . 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 2762 . . . . . . . . 9 𝑢𝐾
50 nfcv 2762 . . . . . . . . 9 𝑣𝐾
51 nfcv 2762 . . . . . . . . 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 2762 . . . . . . . . 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 6643 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (dom 𝑓 +𝑜 𝑥) = (dom 𝑓 +𝑜 𝑦))
5554cbvmptv 4741 . . . . . . . . . . . 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 6182 . . . . . . . . . . . . . . . 16 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝐺𝑘) = (𝐺𝑢))
5958oveq2d 6651 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → (ω ↑𝑜 (𝐺𝑘)) = (ω ↑𝑜 (𝐺𝑢)))
6058fveq2d 6182 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝐹‘(𝐺𝑘)) = (𝐹‘(𝐺𝑢)))
6159, 60oveq12d 6653 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) = ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))))
6256, 61syl5eq 2666 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑀 = ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))))
63 simpr 477 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑓 = 𝑣)
6463dmeqd 5315 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → dom 𝑓 = dom 𝑣)
6564oveq1d 6650 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → (dom 𝑓 +𝑜 𝑦) = (dom 𝑣 +𝑜 𝑦))
6662, 65mpteq12dv 4724 . . . . . . . . . . . 12 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑦𝑀 ↦ (dom 𝑓 +𝑜 𝑦)) = (𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)))
6755, 66syl5eq 2666 . . . . . . . . . . 11 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) = (𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)))
68 oveq2 6643 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑀 +𝑜 𝑥) = (𝑀 +𝑜 𝑦))
6968cbvmptv 4741 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑦))
7062oveq1d 6650 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑀 +𝑜 𝑦) = (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦))
7164, 70mpteq12dv 4724 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑦 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑦)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
7269, 71syl5eq 2666 . . . . . . . . . . . 12 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
7372cnveqd 5287 . . . . . . . . . . 11 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
7467, 73uneq12d 3760 . . . . . . . . . 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 2666 . . . . . . . . 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 6719 . . . . . . . 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 6182 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝐺𝑢) = (𝐺𝐼))
8079oveq2d 6651 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (ω ↑𝑜 (𝐺𝑢)) = (ω ↑𝑜 (𝐺𝐼)))
8179fveq2d 6182 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝐹‘(𝐺𝑢)) = (𝐹‘(𝐺𝐼)))
8280, 81oveq12d 6653 . . . . . . . . 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 5315 . . . . . . . . . . 11 ((𝑢 = 𝐼𝑣 = (𝑇𝐼)) → dom 𝑣 = dom (𝑇𝐼))
85 cnfcom.3 . . . . . . . . . . . 12 (𝜑 → (𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂)
86 f1odm 6128 . . . . . . . . . . . 12 ((𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂 → dom (𝑇𝐼) = (𝐻𝐼))
8785, 86syl 17 . . . . . . . . . . 11 (𝜑 → dom (𝑇𝐼) = (𝐻𝐼))
8884, 87sylan9eqr 2676 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → dom 𝑣 = (𝐻𝐼))
8988oveq1d 6650 . . . . . . . . 9 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (dom 𝑣 +𝑜 𝑦) = ((𝐻𝐼) +𝑜 𝑦))
9082, 89mpteq12dv 4724 . . . . . . . 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 6650 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))
9288, 91mpteq12dv 4724 . . . . . . . . 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 5287 . . . . . . . 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 3760 . . . . . . 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 3207 . . . . . . . 8 (𝐼 ∈ dom 𝐺𝐼 ∈ V)
9620, 95syl 17 . . . . . . 7 (𝜑𝐼 ∈ V)
97 fvexd 6190 . . . . . . 7 (𝜑 → (𝑇𝐼) ∈ V)
98 ovex 6663 . . . . . . . . . 10 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ V
9998mptex 6471 . . . . . . . . 9 (𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∈ V
100 fvex 6188 . . . . . . . . . . 11 (𝐻𝐼) ∈ V
101100mptex 6471 . . . . . . . . . 10 (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)) ∈ V
102101cnvex 7098 . . . . . . . . 9 (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)) ∈ V
10399, 102unex 6941 . . . . . . . 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 6773 . . . . . 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 2654 . . . . 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 6114 . . . . 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 473 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → 𝐴 ∈ On)
112 simpr 477 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → 𝐹𝑆)
11356oveq1i 6645 . . . . . . . . . 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 6705 . . . . . . . 8 (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
116 eqid 2620 . . . . . . . 8 ∅ = ∅
117 seqomeq12 7534 . . . . . . . 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 707 . . . . . . 7 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
11939, 118eqtri 2642 . . . . . 6 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
1205, 110, 111, 21, 112, 119cantnfsuc 8552 . . . . 5 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ 𝐼 ∈ ω) → (𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼)))
1212, 13, 38, 120syl21anc 1323 . . . 4 (𝜑 → (𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼)))
122 f1oeq2 6115 . . . 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 5790 . . . . . 6 dom 𝐺 ⊆ suc dom 𝐺
126125, 20sseldi 3593 . . . . 5 (𝜑𝐼 ∈ suc dom 𝐺)
127 epelg 5020 . . . . . . . . . . 11 (𝐼 ∈ dom 𝐺 → (𝑦 E 𝐼𝑦𝐼))
12820, 127syl 17 . . . . . . . . . 10 (𝜑 → (𝑦 E 𝐼𝑦𝐼))
129128biimpar 502 . . . . . . . . 9 ((𝜑𝑦𝐼) → 𝑦 E 𝐼)
130 ovexd 6665 . . . . . . . . . . . 12 (𝜑 → (𝐹 supp ∅) ∈ V)
13135simpld 475 . . . . . . . . . . . 12 (𝜑 → E We (𝐹 supp ∅))
13221oiiso 8427 . . . . . . . . . . . 12 (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
133130, 131, 132syl2anc 692 . . . . . . . . . . 11 (𝜑𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
134133adantr 481 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
13521oicl 8419 . . . . . . . . . . . 12 Ord dom 𝐺
136 ordelss 5727 . . . . . . . . . . . 12 ((Ord dom 𝐺𝐼 ∈ dom 𝐺) → 𝐼 ⊆ dom 𝐺)
137135, 20, 136sylancr 694 . . . . . . . . . . 11 (𝜑𝐼 ⊆ dom 𝐺)
138137sselda 3595 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝑦 ∈ dom 𝐺)
13920adantr 481 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐼 ∈ dom 𝐺)
140 isorel 6561 . . . . . . . . . 10 ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) ∧ (𝑦 ∈ dom 𝐺𝐼 ∈ dom 𝐺)) → (𝑦 E 𝐼 ↔ (𝐺𝑦) E (𝐺𝐼)))
141134, 138, 139, 140syl12anc 1322 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑦 E 𝐼 ↔ (𝐺𝑦) E (𝐺𝐼)))
142129, 141mpbid 222 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝐺𝑦) E (𝐺𝐼))
143 fvex 6188 . . . . . . . . 9 (𝐺𝐼) ∈ V
144143epelc 5021 . . . . . . . 8 ((𝐺𝑦) E (𝐺𝐼) ↔ (𝐺𝑦) ∈ (𝐺𝐼))
145142, 144sylib 208 . . . . . . 7 ((𝜑𝑦𝐼) → (𝐺𝑦) ∈ (𝐺𝐼))
146145ralrimiva 2963 . . . . . 6 (𝜑 → ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼))
147 ffun 6035 . . . . . . . 8 (𝐺:dom 𝐺⟶(𝐹 supp ∅) → Fun 𝐺)
14822, 147ax-mp 5 . . . . . . 7 Fun 𝐺
149 funimass4 6234 . . . . . . 7 ((Fun 𝐺𝐼 ⊆ dom 𝐺) → ((𝐺𝐼) ⊆ (𝐺𝐼) ↔ ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼)))
150148, 137, 149sylancr 694 . . . . . 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 789 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐴 ∈ On)
154 simplr 791 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐹𝑆)
155 peano1 7070 . . . . . . 7 ∅ ∈ ω
156155a1i 11 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → ∅ ∈ ω)
157 simpr1 1065 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐼 ∈ suc dom 𝐺)
158 simpr2 1066 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐺𝐼) ∈ On)
159 simpr3 1067 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐺𝐼) ⊆ (𝐺𝐼))
1605, 152, 153, 21, 154, 119, 156, 157, 158, 159cantnflt 8554 . . . . 5 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐻𝐼) ∈ (ω ↑𝑜 (𝐺𝐼)))
1612, 13, 126, 27, 151, 160syl23anc 1331 . . . 4 (𝜑 → (𝐻𝐼) ∈ (ω ↑𝑜 (𝐺𝐼)))
162 ffn 6032 . . . . . . . . . 10 (𝐹:𝐴⟶ω → 𝐹 Fn 𝐴)
16316, 162syl 17 . . . . . . . . 9 (𝜑𝐹 Fn 𝐴)
164 0ex 4781 . . . . . . . . . 10 ∅ ∈ V
165164a1i 11 . . . . . . . . 9 (𝜑 → ∅ ∈ V)
166 elsuppfn 7288 . . . . . . . . 9 ((𝐹 Fn 𝐴𝐴 ∈ On ∧ ∅ ∈ V) → ((𝐺𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅)))
167163, 2, 165, 166syl3anc 1324 . . . . . . . 8 (𝜑 → ((𝐺𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅)))
168 simpr 477 . . . . . . . 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 5768 . . . . . . 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 7638 . . . . 5 ((((ω ↑𝑜 (𝐺𝐼)) ∈ On ∧ (𝐹‘(𝐺𝐼)) ∈ On) ∧ ∅ ∈ (𝐹‘(𝐺𝐼))) → (ω ↑𝑜 (𝐺𝐼)) ⊆ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
17529, 32, 173, 174syl21anc 1323 . . . 4 (𝜑 → (ω ↑𝑜 (𝐺𝐼)) ⊆ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
176 oaabs2 7710 . . . 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 1323 . . 3 (𝜑 → ((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
178 f1oeq3 6116 . . 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 384   ∧ w3a 1036   = wceq 1481   ∈ wcel 1988   ≠ wne 2791  ∀wral 2909  Vcvv 3195   ∪ cun 3565   ⊆ wss 3567  ∅c0 3907   class class class wbr 4644   ↦ cmpt 4720   E cep 5018   We wwe 5062  ◡ccnv 5103  dom cdm 5104   “ cima 5107  Ord word 5710  Oncon0 5711  suc csuc 5713  Fun wfun 5870   Fn wfn 5871  ⟶wf 5872  –1-1-onto→wf1o 5875  ‘cfv 5876   Isom wiso 5877  (class class class)co 6635   ↦ cmpt2 6637  ωcom 7050   supp csupp 7280  seq𝜔cseqom 7527   +𝑜 coa 7542   ·𝑜 comu 7543   ↑𝑜 coe 7544   finSupp cfsupp 8260  OrdIsocoi 8399   CNF ccnf 8543 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seqom 7528  df-1o 7545  df-2o 7546  df-oadd 7549  df-omul 7550  df-oexp 7551  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-oi 8400  df-cnf 8544 This theorem is referenced by:  cnfcom  8582
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