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Theorem cantnflem1d 8937
Description: Lemma for cantnf 8942. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
oemapval.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
oemapval.f (𝜑𝐹𝑆)
oemapval.g (𝜑𝐺𝑆)
oemapvali.r (𝜑𝐹𝑇𝐺)
oemapvali.x 𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}
cantnflem1.o 𝑂 = OrdIso( E , (𝐺 supp ∅))
cantnflem1.h 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑂𝑘)) ·o (𝐺‘(𝑂𝑘))) +o 𝑧)), ∅)
Assertion
Ref Expression
cantnflem1d (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) ∈ (𝐻‘suc (𝑂𝑋)))
Distinct variable groups:   𝑘,𝑐,𝑤,𝑥,𝑦,𝑧,𝐵   𝐴,𝑐,𝑘,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐,𝑘   𝑘,𝐹,𝑤,𝑥,𝑦,𝑧   𝑆,𝑐,𝑘,𝑥,𝑦,𝑧   𝐺,𝑐,𝑘,𝑤,𝑥,𝑦,𝑧   𝑥,𝐻,𝑦   𝑘,𝑂,𝑤,𝑥,𝑦,𝑧   𝜑,𝑘,𝑥,𝑦,𝑧   𝑘,𝑋,𝑤,𝑥,𝑦,𝑧   𝐹,𝑐   𝜑,𝑐
Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝐻(𝑧,𝑤,𝑘,𝑐)   𝑂(𝑐)   𝑋(𝑐)

Proof of Theorem cantnflem1d
StepHypRef Expression
1 cantnfs.a . . . . . 6 (𝜑𝐴 ∈ On)
2 cantnfs.b . . . . . . 7 (𝜑𝐵 ∈ On)
3 cantnfs.s . . . . . . . . 9 𝑆 = dom (𝐴 CNF 𝐵)
4 oemapval.t . . . . . . . . 9 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
5 oemapval.f . . . . . . . . 9 (𝜑𝐹𝑆)
6 oemapval.g . . . . . . . . 9 (𝜑𝐺𝑆)
7 oemapvali.r . . . . . . . . 9 (𝜑𝐹𝑇𝐺)
8 oemapvali.x . . . . . . . . 9 𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}
93, 1, 2, 4, 5, 6, 7, 8oemapvali 8933 . . . . . . . 8 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
109simp1d 1122 . . . . . . 7 (𝜑𝑋𝐵)
11 onelon 6048 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
122, 10, 11syl2anc 576 . . . . . 6 (𝜑𝑋 ∈ On)
13 oecl 7956 . . . . . 6 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
141, 12, 13syl2anc 576 . . . . 5 (𝜑 → (𝐴o 𝑋) ∈ On)
153, 1, 2cantnfs 8915 . . . . . . . . 9 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
166, 15mpbid 224 . . . . . . . 8 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1716simpld 487 . . . . . . 7 (𝜑𝐺:𝐵𝐴)
1817, 10ffvelrnd 6671 . . . . . 6 (𝜑 → (𝐺𝑋) ∈ 𝐴)
19 onelon 6048 . . . . . 6 ((𝐴 ∈ On ∧ (𝐺𝑋) ∈ 𝐴) → (𝐺𝑋) ∈ On)
201, 18, 19syl2anc 576 . . . . 5 (𝜑 → (𝐺𝑋) ∈ On)
21 omcl 7955 . . . . 5 (((𝐴o 𝑋) ∈ On ∧ (𝐺𝑋) ∈ On) → ((𝐴o 𝑋) ·o (𝐺𝑋)) ∈ On)
2214, 20, 21syl2anc 576 . . . 4 (𝜑 → ((𝐴o 𝑋) ·o (𝐺𝑋)) ∈ On)
23 ovexd 7004 . . . . . . . . . 10 (𝜑 → (𝐺 supp ∅) ∈ V)
24 cantnflem1.o . . . . . . . . . . . 12 𝑂 = OrdIso( E , (𝐺 supp ∅))
253, 1, 2, 24, 6cantnfcl 8916 . . . . . . . . . . 11 (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
2625simpld 487 . . . . . . . . . 10 (𝜑 → E We (𝐺 supp ∅))
2724oiiso 8788 . . . . . . . . . 10 (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
2823, 26, 27syl2anc 576 . . . . . . . . 9 (𝜑𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
29 isof1o 6893 . . . . . . . . 9 (𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) → 𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅))
3028, 29syl 17 . . . . . . . 8 (𝜑𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅))
31 f1ocnv 6450 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) → 𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
32 f1of 6438 . . . . . . . 8 (𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂𝑂:(𝐺 supp ∅)⟶dom 𝑂)
3330, 31, 323syl 18 . . . . . . 7 (𝜑𝑂:(𝐺 supp ∅)⟶dom 𝑂)
343, 1, 2, 4, 5, 6, 7, 8cantnflem1a 8934 . . . . . . 7 (𝜑𝑋 ∈ (𝐺 supp ∅))
3533, 34ffvelrnd 6671 . . . . . 6 (𝜑 → (𝑂𝑋) ∈ dom 𝑂)
3625simprd 488 . . . . . 6 (𝜑 → dom 𝑂 ∈ ω)
37 elnn 7400 . . . . . 6 (((𝑂𝑋) ∈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → (𝑂𝑋) ∈ ω)
3835, 36, 37syl2anc 576 . . . . 5 (𝜑 → (𝑂𝑋) ∈ ω)
39 cantnflem1.h . . . . . . 7 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑂𝑘)) ·o (𝐺‘(𝑂𝑘))) +o 𝑧)), ∅)
4039cantnfvalf 8914 . . . . . 6 𝐻:ω⟶On
4140ffvelrni 6669 . . . . 5 ((𝑂𝑋) ∈ ω → (𝐻‘(𝑂𝑋)) ∈ On)
4238, 41syl 17 . . . 4 (𝜑 → (𝐻‘(𝑂𝑋)) ∈ On)
43 oaword1 7971 . . . 4 ((((𝐴o 𝑋) ·o (𝐺𝑋)) ∈ On ∧ (𝐻‘(𝑂𝑋)) ∈ On) → ((𝐴o 𝑋) ·o (𝐺𝑋)) ⊆ (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
4422, 42, 43syl2anc 576 . . 3 (𝜑 → ((𝐴o 𝑋) ·o (𝐺𝑋)) ⊆ (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
453, 1, 2, 24, 6, 39cantnfsuc 8919 . . . . 5 ((𝜑 ∧ (𝑂𝑋) ∈ ω) → (𝐻‘suc (𝑂𝑋)) = (((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) +o (𝐻‘(𝑂𝑋))))
4638, 45mpdan 674 . . . 4 (𝜑 → (𝐻‘suc (𝑂𝑋)) = (((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) +o (𝐻‘(𝑂𝑋))))
47 f1ocnvfv2 6853 . . . . . . . 8 ((𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(𝑂𝑋)) = 𝑋)
4830, 34, 47syl2anc 576 . . . . . . 7 (𝜑 → (𝑂‘(𝑂𝑋)) = 𝑋)
4948oveq2d 6986 . . . . . 6 (𝜑 → (𝐴o (𝑂‘(𝑂𝑋))) = (𝐴o 𝑋))
5048fveq2d 6497 . . . . . 6 (𝜑 → (𝐺‘(𝑂‘(𝑂𝑋))) = (𝐺𝑋))
5149, 50oveq12d 6988 . . . . 5 (𝜑 → ((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) = ((𝐴o 𝑋) ·o (𝐺𝑋)))
5251oveq1d 6985 . . . 4 (𝜑 → (((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) +o (𝐻‘(𝑂𝑋))) = (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
5346, 52eqtrd 2808 . . 3 (𝜑 → (𝐻‘suc (𝑂𝑋)) = (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
5444, 53sseqtr4d 3894 . 2 (𝜑 → ((𝐴o 𝑋) ·o (𝐺𝑋)) ⊆ (𝐻‘suc (𝑂𝑋)))
55 onss 7315 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐵 ⊆ On)
562, 55syl 17 . . . . . . . . . 10 (𝜑𝐵 ⊆ On)
5756sselda 3854 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑥 ∈ On)
5812adantr 473 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑋 ∈ On)
59 onsseleq 6064 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
6057, 58, 59syl2anc 576 . . . . . . . 8 ((𝜑𝑥𝐵) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
61 orcom 856 . . . . . . . 8 ((𝑥𝑋𝑥 = 𝑋) ↔ (𝑥 = 𝑋𝑥𝑋))
6260, 61syl6bb 279 . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥𝑋 ↔ (𝑥 = 𝑋𝑥𝑋)))
6362ifbid 4366 . . . . . 6 ((𝜑𝑥𝐵) → if(𝑥𝑋, (𝐹𝑥), ∅) = if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))
6463mpteq2dva 5016 . . . . 5 (𝜑 → (𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅)) = (𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)))
6564fveq2d 6497 . . . 4 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) = ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))))
663, 1, 2cantnfs 8915 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐵𝐴𝐹 finSupp ∅)))
675, 66mpbid 224 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐵𝐴𝐹 finSupp ∅))
6867simpld 487 . . . . . . . . . 10 (𝜑𝐹:𝐵𝐴)
6968ffvelrnda 6670 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝐹𝑦) ∈ 𝐴)
7018ne0d 4182 . . . . . . . . . . 11 (𝜑𝐴 ≠ ∅)
71 on0eln0 6078 . . . . . . . . . . . 12 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
721, 71syl 17 . . . . . . . . . . 11 (𝜑 → (∅ ∈ 𝐴𝐴 ≠ ∅))
7370, 72mpbird 249 . . . . . . . . . 10 (𝜑 → ∅ ∈ 𝐴)
7473adantr 473 . . . . . . . . 9 ((𝜑𝑦𝐵) → ∅ ∈ 𝐴)
7569, 74ifcld 4389 . . . . . . . 8 ((𝜑𝑦𝐵) → if(𝑦𝑋, (𝐹𝑦), ∅) ∈ 𝐴)
7675fmpttd 6696 . . . . . . 7 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)):𝐵𝐴)
77 0ex 5062 . . . . . . . . 9 ∅ ∈ V
7877a1i 11 . . . . . . . 8 (𝜑 → ∅ ∈ V)
7967simprd 488 . . . . . . . 8 (𝜑𝐹 finSupp ∅)
8068, 2, 78, 79fsuppmptif 8650 . . . . . . 7 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) finSupp ∅)
813, 1, 2cantnfs 8915 . . . . . . 7 (𝜑 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) ∈ 𝑆 ↔ ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)):𝐵𝐴 ∧ (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) finSupp ∅)))
8276, 80, 81mpbir2and 700 . . . . . 6 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) ∈ 𝑆)
8368, 10ffvelrnd 6671 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ 𝐴)
84 eldifn 3990 . . . . . . . . 9 (𝑦 ∈ (𝐵𝑋) → ¬ 𝑦𝑋)
8584adantl 474 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐵𝑋)) → ¬ 𝑦𝑋)
8685iffalsed 4355 . . . . . . 7 ((𝜑𝑦 ∈ (𝐵𝑋)) → if(𝑦𝑋, (𝐹𝑦), ∅) = ∅)
8786, 2suppss2 7660 . . . . . 6 (𝜑 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) supp ∅) ⊆ 𝑋)
88 fveq2 6493 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
8988adantl 474 . . . . . . . . . 10 ((𝑥𝐵𝑥 = 𝑋) → (𝐹𝑥) = (𝐹𝑋))
9089ifeq1da 4374 . . . . . . . . 9 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑥), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
91 eleq1w 2842 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦𝑋𝑥𝑋))
92 fveq2 6493 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
9391, 92ifbieq1d 4367 . . . . . . . . . . 11 (𝑦 = 𝑥 → if(𝑦𝑋, (𝐹𝑦), ∅) = if(𝑥𝑋, (𝐹𝑥), ∅))
94 eqid 2772 . . . . . . . . . . 11 (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) = (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))
95 fvex 6506 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
9695, 77ifex 4392 . . . . . . . . . . 11 if(𝑥𝑋, (𝐹𝑥), ∅) ∈ V
9793, 94, 96fvmpt 6589 . . . . . . . . . 10 (𝑥𝐵 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥) = if(𝑥𝑋, (𝐹𝑥), ∅))
9897ifeq2d 4363 . . . . . . . . 9 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑥), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅)))
9990, 98eqtr3d 2810 . . . . . . . 8 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅)))
100 ifor 4396 . . . . . . . 8 if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅))
10199, 100syl6reqr 2827 . . . . . . 7 (𝑥𝐵 → if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅) = if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
102101mpteq2ia 5012 . . . . . 6 (𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)) = (𝑥𝐵 ↦ if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
1033, 1, 2, 82, 10, 83, 87, 102cantnfp1 8930 . . . . 5 (𝜑 → ((𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)) ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))))))
104103simprd 488 . . . 4 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))))
10565, 104eqtrd 2808 . . 3 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))))
106 onelon 6048 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐹𝑋) ∈ 𝐴) → (𝐹𝑋) ∈ On)
1071, 83, 106syl2anc 576 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ On)
108 omsuc 7945 . . . . . 6 (((𝐴o 𝑋) ∈ On ∧ (𝐹𝑋) ∈ On) → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)))
10914, 107, 108syl2anc 576 . . . . 5 (𝜑 → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)))
110 eloni 6033 . . . . . . . 8 ((𝐺𝑋) ∈ On → Ord (𝐺𝑋))
11120, 110syl 17 . . . . . . 7 (𝜑 → Ord (𝐺𝑋))
1129simp2d 1123 . . . . . . 7 (𝜑 → (𝐹𝑋) ∈ (𝐺𝑋))
113 ordsucss 7343 . . . . . . 7 (Ord (𝐺𝑋) → ((𝐹𝑋) ∈ (𝐺𝑋) → suc (𝐹𝑋) ⊆ (𝐺𝑋)))
114111, 112, 113sylc 65 . . . . . 6 (𝜑 → suc (𝐹𝑋) ⊆ (𝐺𝑋))
115 suceloni 7338 . . . . . . . 8 ((𝐹𝑋) ∈ On → suc (𝐹𝑋) ∈ On)
116107, 115syl 17 . . . . . . 7 (𝜑 → suc (𝐹𝑋) ∈ On)
117 omwordi 7990 . . . . . . 7 ((suc (𝐹𝑋) ∈ On ∧ (𝐺𝑋) ∈ On ∧ (𝐴o 𝑋) ∈ On) → (suc (𝐹𝑋) ⊆ (𝐺𝑋) → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) ⊆ ((𝐴o 𝑋) ·o (𝐺𝑋))))
118116, 20, 14, 117syl3anc 1351 . . . . . 6 (𝜑 → (suc (𝐹𝑋) ⊆ (𝐺𝑋) → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) ⊆ ((𝐴o 𝑋) ·o (𝐺𝑋))))
119114, 118mpd 15 . . . . 5 (𝜑 → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) ⊆ ((𝐴o 𝑋) ·o (𝐺𝑋)))
120109, 119eqsstr3d 3892 . . . 4 (𝜑 → (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)) ⊆ ((𝐴o 𝑋) ·o (𝐺𝑋)))
1213, 1, 2, 82, 73, 12, 87cantnflt2 8922 . . . . 5 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴o 𝑋))
122 onelon 6048 . . . . . . 7 (((𝐴o 𝑋) ∈ On ∧ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴o 𝑋)) → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On)
12314, 121, 122syl2anc 576 . . . . . 6 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On)
124 omcl 7955 . . . . . . 7 (((𝐴o 𝑋) ∈ On ∧ (𝐹𝑋) ∈ On) → ((𝐴o 𝑋) ·o (𝐹𝑋)) ∈ On)
12514, 107, 124syl2anc 576 . . . . . 6 (𝜑 → ((𝐴o 𝑋) ·o (𝐹𝑋)) ∈ On)
126 oaord 7966 . . . . . 6 ((((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On ∧ (𝐴o 𝑋) ∈ On ∧ ((𝐴o 𝑋) ·o (𝐹𝑋)) ∈ On) → (((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴o 𝑋) ↔ (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋))))
127123, 14, 125, 126syl3anc 1351 . . . . 5 (𝜑 → (((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴o 𝑋) ↔ (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋))))
128121, 127mpbid 224 . . . 4 (𝜑 → (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)))
129120, 128sseldd 3855 . . 3 (𝜑 → (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ ((𝐴o 𝑋) ·o (𝐺𝑋)))
130105, 129eqeltrd 2860 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) ∈ ((𝐴o 𝑋) ·o (𝐺𝑋)))
13154, 130sseldd 3855 1 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) ∈ (𝐻‘suc (𝑂𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wo 833   = wceq 1507  wcel 2048  wne 2961  wral 3082  wrex 3083  {crab 3086  Vcvv 3409  cdif 3822  wss 3825  c0 4173  ifcif 4344   cuni 4706   class class class wbr 4923  {copab 4985  cmpt 5002   E cep 5309   We wwe 5358  ccnv 5399  dom cdm 5400  Ord word 6022  Oncon0 6023  suc csuc 6025  wf 6178  1-1-ontowf1o 6181  cfv 6182   Isom wiso 6183  (class class class)co 6970  cmpo 6972  ωcom 7390   supp csupp 7626  seq𝜔cseqom 7879   +o coa 7894   ·o comu 7895  o coe 7896   finSupp cfsupp 8620  OrdIsocoi 8760   CNF ccnf 8910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-supp 7627  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-seqom 7880  df-1o 7897  df-2o 7898  df-oadd 7901  df-omul 7902  df-oexp 7903  df-er 8081  df-map 8200  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-fsupp 8621  df-oi 8761  df-cnf 8911
This theorem is referenced by:  cantnflem1  8938
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