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Theorem cnfcomlem 9314
Description: Lemma for cnfcom 9315. (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 9261 . . . . . . 7 ω ∈ On
2 cnfcom.a . . . . . . . 8 (𝜑𝐴 ∈ On)
3 suppssdm 7919 . . . . . . . . . 10 (𝐹 supp ∅) ⊆ dom 𝐹
4 cnfcom.f . . . . . . . . . . . . 13 𝐹 = ((ω CNF 𝐴)‘𝐵)
5 cnfcom.s . . . . . . . . . . . . . . . 16 𝑆 = dom (ω CNF 𝐴)
61a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → ω ∈ On)
75, 6, 2cantnff1o 9311 . . . . . . . . . . . . . . 15 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑o 𝐴))
8 f1ocnv 6673 . . . . . . . . . . . . . . 15 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑o 𝐴) → (ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto𝑆)
9 f1of 6661 . . . . . . . . . . . . . . 15 ((ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto𝑆(ω CNF 𝐴):(ω ↑o 𝐴)⟶𝑆)
107, 8, 93syl 18 . . . . . . . . . . . . . 14 (𝜑(ω CNF 𝐴):(ω ↑o 𝐴)⟶𝑆)
11 cnfcom.b . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ (ω ↑o 𝐴))
1210, 11ffvelrnd 6905 . . . . . . . . . . . . 13 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
134, 12eqeltrid 2842 . . . . . . . . . . . 12 (𝜑𝐹𝑆)
145, 6, 2cantnfs 9281 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
1513, 14mpbid 235 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
1615simpld 498 . . . . . . . . . 10 (𝜑𝐹:𝐴⟶ω)
173, 16fssdm 6565 . . . . . . . . 9 (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
18 cnfcom.1 . . . . . . . . . 10 (𝜑𝐼 ∈ dom 𝐺)
19 cnfcom.g . . . . . . . . . . . 12 𝐺 = OrdIso( E , (𝐹 supp ∅))
2019oif 9146 . . . . . . . . . . 11 𝐺:dom 𝐺⟶(𝐹 supp ∅)
2120ffvelrni 6903 . . . . . . . . . 10 (𝐼 ∈ dom 𝐺 → (𝐺𝐼) ∈ (𝐹 supp ∅))
2218, 21syl 17 . . . . . . . . 9 (𝜑 → (𝐺𝐼) ∈ (𝐹 supp ∅))
2317, 22sseldd 3902 . . . . . . . 8 (𝜑 → (𝐺𝐼) ∈ 𝐴)
24 onelon 6238 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐺𝐼) ∈ 𝐴) → (𝐺𝐼) ∈ On)
252, 23, 24syl2anc 587 . . . . . . 7 (𝜑 → (𝐺𝐼) ∈ On)
26 oecl 8264 . . . . . . 7 ((ω ∈ On ∧ (𝐺𝐼) ∈ On) → (ω ↑o (𝐺𝐼)) ∈ On)
271, 25, 26sylancr 590 . . . . . 6 (𝜑 → (ω ↑o (𝐺𝐼)) ∈ On)
2816, 23ffvelrnd 6905 . . . . . . 7 (𝜑 → (𝐹‘(𝐺𝐼)) ∈ ω)
29 nnon 7650 . . . . . . 7 ((𝐹‘(𝐺𝐼)) ∈ ω → (𝐹‘(𝐺𝐼)) ∈ On)
3028, 29syl 17 . . . . . 6 (𝜑 → (𝐹‘(𝐺𝐼)) ∈ On)
31 omcl 8263 . . . . . 6 (((ω ↑o (𝐺𝐼)) ∈ On ∧ (𝐹‘(𝐺𝐼)) ∈ On) → ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) ∈ On)
3227, 30, 31syl2anc 587 . . . . 5 (𝜑 → ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) ∈ On)
335, 6, 2, 19, 13cantnfcl 9282 . . . . . . . 8 (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
3433simprd 499 . . . . . . 7 (𝜑 → dom 𝐺 ∈ ω)
35 elnn 7655 . . . . . . 7 ((𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω) → 𝐼 ∈ ω)
3618, 34, 35syl2anc 587 . . . . . 6 (𝜑𝐼 ∈ ω)
37 cnfcom.h . . . . . . . 8 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)
3837cantnfvalf 9280 . . . . . . 7 𝐻:ω⟶On
3938ffvelrni 6903 . . . . . 6 (𝐼 ∈ ω → (𝐻𝐼) ∈ On)
4036, 39syl 17 . . . . 5 (𝜑 → (𝐻𝐼) ∈ On)
41 eqid 2737 . . . . . 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)))
4241oacomf1o 8293 . . . . 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)))))
4332, 40, 42syl2anc 587 . . . 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)))))
44 cnfcom.t . . . . . . . 8 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
4544seqomsuc 8193 . . . . . . 7 (𝐼 ∈ ω → (𝑇‘suc 𝐼) = (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇𝐼)))
4636, 45syl 17 . . . . . 6 (𝜑 → (𝑇‘suc 𝐼) = (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇𝐼)))
47 nfcv 2904 . . . . . . . . 9 𝑢𝐾
48 nfcv 2904 . . . . . . . . 9 𝑣𝐾
49 nfcv 2904 . . . . . . . . 9 𝑘((𝑦 ∈ ((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +o 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑦)))
50 nfcv 2904 . . . . . . . . 9 𝑓((𝑦 ∈ ((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +o 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑦)))
51 cnfcom.k . . . . . . . . . 10 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))
52 oveq2 7221 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (dom 𝑓 +o 𝑥) = (dom 𝑓 +o 𝑦))
5352cbvmptv 5158 . . . . . . . . . . . 12 (𝑥𝑀 ↦ (dom 𝑓 +o 𝑥)) = (𝑦𝑀 ↦ (dom 𝑓 +o 𝑦))
54 cnfcom.m . . . . . . . . . . . . . 14 𝑀 = ((ω ↑o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘)))
55 simpl 486 . . . . . . . . . . . . . . . . 17 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑘 = 𝑢)
5655fveq2d 6721 . . . . . . . . . . . . . . . 16 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝐺𝑘) = (𝐺𝑢))
5756oveq2d 7229 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → (ω ↑o (𝐺𝑘)) = (ω ↑o (𝐺𝑢)))
5856fveq2d 6721 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝐹‘(𝐺𝑘)) = (𝐹‘(𝐺𝑢)))
5957, 58oveq12d 7231 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → ((ω ↑o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) = ((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))))
6054, 59eqtrid 2789 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑀 = ((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))))
61 simpr 488 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑓 = 𝑣)
6261dmeqd 5774 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → dom 𝑓 = dom 𝑣)
6362oveq1d 7228 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → (dom 𝑓 +o 𝑦) = (dom 𝑣 +o 𝑦))
6460, 63mpteq12dv 5140 . . . . . . . . . . . 12 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑦𝑀 ↦ (dom 𝑓 +o 𝑦)) = (𝑦 ∈ ((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +o 𝑦)))
6553, 64eqtrid 2789 . . . . . . . . . . 11 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥𝑀 ↦ (dom 𝑓 +o 𝑥)) = (𝑦 ∈ ((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +o 𝑦)))
66 oveq2 7221 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑀 +o 𝑥) = (𝑀 +o 𝑦))
6766cbvmptv 5158 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)) = (𝑦 ∈ dom 𝑓 ↦ (𝑀 +o 𝑦))
6860oveq1d 7228 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑀 +o 𝑦) = (((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑦))
6962, 68mpteq12dv 5140 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑦 ∈ dom 𝑓 ↦ (𝑀 +o 𝑦)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑦)))
7067, 69eqtrid 2789 . . . . . . . . . . . 12 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑦)))
7170cnveqd 5744 . . . . . . . . . . 11 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑦)))
7265, 71uneq12d 4078 . . . . . . . . . 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))))
7351, 72eqtrid 2789 . . . . . . . . 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))))
7447, 48, 49, 50, 73cbvmpo 7305 . . . . . . . 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))))
7574a1i 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)))))
76 simprl 771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → 𝑢 = 𝐼)
7776fveq2d 6721 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝐺𝑢) = (𝐺𝐼))
7877oveq2d 7229 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (ω ↑o (𝐺𝑢)) = (ω ↑o (𝐺𝐼)))
7977fveq2d 6721 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝐹‘(𝐺𝑢)) = (𝐹‘(𝐺𝐼)))
8078, 79oveq12d 7231 . . . . . . . . 9 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → ((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) = ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))))
81 simpr 488 . . . . . . . . . . . 12 ((𝑢 = 𝐼𝑣 = (𝑇𝐼)) → 𝑣 = (𝑇𝐼))
8281dmeqd 5774 . . . . . . . . . . 11 ((𝑢 = 𝐼𝑣 = (𝑇𝐼)) → dom 𝑣 = dom (𝑇𝐼))
83 cnfcom.3 . . . . . . . . . . . 12 (𝜑 → (𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂)
84 f1odm 6665 . . . . . . . . . . . 12 ((𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂 → dom (𝑇𝐼) = (𝐻𝐼))
8583, 84syl 17 . . . . . . . . . . 11 (𝜑 → dom (𝑇𝐼) = (𝐻𝐼))
8682, 85sylan9eqr 2800 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → dom 𝑣 = (𝐻𝐼))
8786oveq1d 7228 . . . . . . . . 9 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (dom 𝑣 +o 𝑦) = ((𝐻𝐼) +o 𝑦))
8880, 87mpteq12dv 5140 . . . . . . . 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)))
8980oveq1d 7228 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (((ω ↑o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑦) = (((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) +o 𝑦))
9086, 89mpteq12dv 5140 . . . . . . . . 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)))
9190cnveqd 5744 . . . . . . . 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)))
9288, 91uneq12d 4078 . . . . . . 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))))
9318elexd 3428 . . . . . . 7 (𝜑𝐼 ∈ V)
94 fvexd 6732 . . . . . . 7 (𝜑 → (𝑇𝐼) ∈ V)
95 ovex 7246 . . . . . . . . . 10 ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) ∈ V
9695mptex 7039 . . . . . . . . 9 (𝑦 ∈ ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +o 𝑦)) ∈ V
97 fvex 6730 . . . . . . . . . . 11 (𝐻𝐼) ∈ V
9897mptex 7039 . . . . . . . . . 10 (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) +o 𝑦)) ∈ V
9998cnvex 7703 . . . . . . . . 9 (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) +o 𝑦)) ∈ V
10096, 99unex 7531 . . . . . . . 8 ((𝑦 ∈ ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +o 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) +o 𝑦))) ∈ V
101100a1i 11 . . . . . . 7 (𝜑 → ((𝑦 ∈ ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +o 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) +o 𝑦))) ∈ V)
10275, 92, 93, 94, 101ovmpod 7361 . . . . . 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))))
10346, 102eqtrd 2777 . . . . 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))))
104103f1oeq1d 6656 . . . 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))))))
10543, 104mpbird 260 . . 3 (𝜑 → (𝑇‘suc 𝐼):(((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) +o (𝐻𝐼))–1-1-onto→((𝐻𝐼) +o ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼)))))
1061a1i 11 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → ω ∈ On)
107 simpl 486 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → 𝐴 ∈ On)
108 simpr 488 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → 𝐹𝑆)
10954oveq1i 7223 . . . . . . . . . 10 (𝑀 +o 𝑧) = (((ω ↑o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)
110109a1i 11 . . . . . . . . 9 ((𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑀 +o 𝑧) = (((ω ↑o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))
111110mpoeq3ia 7289 . . . . . . . 8 (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))
112 eqid 2737 . . . . . . . 8 ∅ = ∅
113 seqomeq12 8190 . . . . . . . 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)), ∅))
114111, 112, 113mp2an 692 . . . . . . 7 seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)
11537, 114eqtri 2765 . . . . . 6 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)
1165, 106, 107, 19, 108, 115cantnfsuc 9285 . . . . 5 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ 𝐼 ∈ ω) → (𝐻‘suc 𝐼) = (((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) +o (𝐻𝐼)))
1172, 13, 36, 116syl21anc 838 . . . 4 (𝜑 → (𝐻‘suc 𝐼) = (((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))) +o (𝐻𝐼)))
118117f1oeq2d 6657 . . 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))))))
119105, 118mpbird 260 . 2 (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻𝐼) +o ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼)))))
120 sssucid 6290 . . . . . 6 dom 𝐺 ⊆ suc dom 𝐺
121120, 18sselid 3898 . . . . 5 (𝜑𝐼 ∈ suc dom 𝐺)
122 epelg 5461 . . . . . . . . . . 11 (𝐼 ∈ dom 𝐺 → (𝑦 E 𝐼𝑦𝐼))
12318, 122syl 17 . . . . . . . . . 10 (𝜑 → (𝑦 E 𝐼𝑦𝐼))
124123biimpar 481 . . . . . . . . 9 ((𝜑𝑦𝐼) → 𝑦 E 𝐼)
125 ovexd 7248 . . . . . . . . . . . 12 (𝜑 → (𝐹 supp ∅) ∈ V)
12633simpld 498 . . . . . . . . . . . 12 (𝜑 → E We (𝐹 supp ∅))
12719oiiso 9153 . . . . . . . . . . . 12 (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
128125, 126, 127syl2anc 587 . . . . . . . . . . 11 (𝜑𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
129128adantr 484 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
13019oicl 9145 . . . . . . . . . . . 12 Ord dom 𝐺
131 ordelss 6229 . . . . . . . . . . . 12 ((Ord dom 𝐺𝐼 ∈ dom 𝐺) → 𝐼 ⊆ dom 𝐺)
132130, 18, 131sylancr 590 . . . . . . . . . . 11 (𝜑𝐼 ⊆ dom 𝐺)
133132sselda 3901 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝑦 ∈ dom 𝐺)
13418adantr 484 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐼 ∈ dom 𝐺)
135 isorel 7135 . . . . . . . . . 10 ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) ∧ (𝑦 ∈ dom 𝐺𝐼 ∈ dom 𝐺)) → (𝑦 E 𝐼 ↔ (𝐺𝑦) E (𝐺𝐼)))
136129, 133, 134, 135syl12anc 837 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑦 E 𝐼 ↔ (𝐺𝑦) E (𝐺𝐼)))
137124, 136mpbid 235 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝐺𝑦) E (𝐺𝐼))
138 fvex 6730 . . . . . . . . 9 (𝐺𝐼) ∈ V
139138epeli 5462 . . . . . . . 8 ((𝐺𝑦) E (𝐺𝐼) ↔ (𝐺𝑦) ∈ (𝐺𝐼))
140137, 139sylib 221 . . . . . . 7 ((𝜑𝑦𝐼) → (𝐺𝑦) ∈ (𝐺𝐼))
141140ralrimiva 3105 . . . . . 6 (𝜑 → ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼))
142 ffun 6548 . . . . . . . 8 (𝐺:dom 𝐺⟶(𝐹 supp ∅) → Fun 𝐺)
14320, 142ax-mp 5 . . . . . . 7 Fun 𝐺
144 funimass4 6777 . . . . . . 7 ((Fun 𝐺𝐼 ⊆ dom 𝐺) → ((𝐺𝐼) ⊆ (𝐺𝐼) ↔ ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼)))
145143, 132, 144sylancr 590 . . . . . 6 (𝜑 → ((𝐺𝐼) ⊆ (𝐺𝐼) ↔ ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼)))
146141, 145mpbird 260 . . . . 5 (𝜑 → (𝐺𝐼) ⊆ (𝐺𝐼))
1471a1i 11 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → ω ∈ On)
148 simpll 767 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐴 ∈ On)
149 simplr 769 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐹𝑆)
150 peano1 7667 . . . . . . 7 ∅ ∈ ω
151150a1i 11 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → ∅ ∈ ω)
152 simpr1 1196 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐼 ∈ suc dom 𝐺)
153 simpr2 1197 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐺𝐼) ∈ On)
154 simpr3 1198 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐺𝐼) ⊆ (𝐺𝐼))
1555, 147, 148, 19, 149, 115, 151, 152, 153, 154cantnflt 9287 . . . . 5 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐻𝐼) ∈ (ω ↑o (𝐺𝐼)))
1562, 13, 121, 25, 146, 155syl23anc 1379 . . . 4 (𝜑 → (𝐻𝐼) ∈ (ω ↑o (𝐺𝐼)))
15716ffnd 6546 . . . . . . . . 9 (𝜑𝐹 Fn 𝐴)
158 0ex 5200 . . . . . . . . . 10 ∅ ∈ V
159158a1i 11 . . . . . . . . 9 (𝜑 → ∅ ∈ V)
160 elsuppfn 7913 . . . . . . . . 9 ((𝐹 Fn 𝐴𝐴 ∈ On ∧ ∅ ∈ V) → ((𝐺𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅)))
161157, 2, 159, 160syl3anc 1373 . . . . . . . 8 (𝜑 → ((𝐺𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅)))
162 simpr 488 . . . . . . . 8 (((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅) → (𝐹‘(𝐺𝐼)) ≠ ∅)
163161, 162syl6bi 256 . . . . . . 7 (𝜑 → ((𝐺𝐼) ∈ (𝐹 supp ∅) → (𝐹‘(𝐺𝐼)) ≠ ∅))
16422, 163mpd 15 . . . . . 6 (𝜑 → (𝐹‘(𝐺𝐼)) ≠ ∅)
165 on0eln0 6268 . . . . . . 7 ((𝐹‘(𝐺𝐼)) ∈ On → (∅ ∈ (𝐹‘(𝐺𝐼)) ↔ (𝐹‘(𝐺𝐼)) ≠ ∅))
16630, 165syl 17 . . . . . 6 (𝜑 → (∅ ∈ (𝐹‘(𝐺𝐼)) ↔ (𝐹‘(𝐺𝐼)) ≠ ∅))
167164, 166mpbird 260 . . . . 5 (𝜑 → ∅ ∈ (𝐹‘(𝐺𝐼)))
168 omword1 8301 . . . . 5 ((((ω ↑o (𝐺𝐼)) ∈ On ∧ (𝐹‘(𝐺𝐼)) ∈ On) ∧ ∅ ∈ (𝐹‘(𝐺𝐼))) → (ω ↑o (𝐺𝐼)) ⊆ ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))))
16927, 30, 167, 168syl21anc 838 . . . 4 (𝜑 → (ω ↑o (𝐺𝐼)) ⊆ ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))))
170 oaabs2 8374 . . . 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))))
171156, 32, 169, 170syl21anc 838 . . 3 (𝜑 → ((𝐻𝐼) +o ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼)))) = ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))))
172171f1oeq3d 6658 . 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)))))
173119, 172mpbid 235 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wral 3061  Vcvv 3408  cun 3864  wss 3866  c0 4237   class class class wbr 5053  cmpt 5135   E cep 5459   We wwe 5508  ccnv 5550  dom cdm 5551  cima 5554  Ord word 6212  Oncon0 6213  suc csuc 6215  Fun wfun 6374   Fn wfn 6375  wf 6376  1-1-ontowf1o 6379  cfv 6380   Isom wiso 6381  (class class class)co 7213  cmpo 7215  ωcom 7644   supp csupp 7903  seqωcseqom 8183   +o coa 8199   ·o comu 8200  o coe 8201   finSupp cfsupp 8985  OrdIsocoi 9125   CNF ccnf 9276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-seqom 8184  df-1o 8202  df-2o 8203  df-oadd 8206  df-omul 8207  df-oexp 8208  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-oi 9126  df-cnf 9277
This theorem is referenced by:  cnfcom  9315
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