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Theorem cantnflem1d 9726
Description: Lemma for cantnf 9731. (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 9722 . . . . . . . 8 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
109simp1d 1141 . . . . . . 7 (𝜑𝑋𝐵)
11 onelon 6411 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
122, 10, 11syl2anc 584 . . . . . 6 (𝜑𝑋 ∈ On)
13 oecl 8574 . . . . . 6 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
141, 12, 13syl2anc 584 . . . . 5 (𝜑 → (𝐴o 𝑋) ∈ On)
153, 1, 2cantnfs 9704 . . . . . . . . 9 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
166, 15mpbid 232 . . . . . . . 8 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1716simpld 494 . . . . . . 7 (𝜑𝐺:𝐵𝐴)
1817, 10ffvelcdmd 7105 . . . . . 6 (𝜑 → (𝐺𝑋) ∈ 𝐴)
19 onelon 6411 . . . . . 6 ((𝐴 ∈ On ∧ (𝐺𝑋) ∈ 𝐴) → (𝐺𝑋) ∈ On)
201, 18, 19syl2anc 584 . . . . 5 (𝜑 → (𝐺𝑋) ∈ On)
21 omcl 8573 . . . . 5 (((𝐴o 𝑋) ∈ On ∧ (𝐺𝑋) ∈ On) → ((𝐴o 𝑋) ·o (𝐺𝑋)) ∈ On)
2214, 20, 21syl2anc 584 . . . 4 (𝜑 → ((𝐴o 𝑋) ·o (𝐺𝑋)) ∈ On)
23 ovexd 7466 . . . . . . . . . 10 (𝜑 → (𝐺 supp ∅) ∈ V)
24 cantnflem1.o . . . . . . . . . . . 12 𝑂 = OrdIso( E , (𝐺 supp ∅))
253, 1, 2, 24, 6cantnfcl 9705 . . . . . . . . . . 11 (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
2625simpld 494 . . . . . . . . . 10 (𝜑 → E We (𝐺 supp ∅))
2724oiiso 9575 . . . . . . . . . 10 (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
2823, 26, 27syl2anc 584 . . . . . . . . 9 (𝜑𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
29 isof1o 7343 . . . . . . . . 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 6861 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) → 𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
32 f1of 6849 . . . . . . . 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 9723 . . . . . . 7 (𝜑𝑋 ∈ (𝐺 supp ∅))
3533, 34ffvelcdmd 7105 . . . . . 6 (𝜑 → (𝑂𝑋) ∈ dom 𝑂)
3625simprd 495 . . . . . 6 (𝜑 → dom 𝑂 ∈ ω)
37 elnn 7898 . . . . . 6 (((𝑂𝑋) ∈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → (𝑂𝑋) ∈ ω)
3835, 36, 37syl2anc 584 . . . . 5 (𝜑 → (𝑂𝑋) ∈ ω)
39 cantnflem1.h . . . . . . 7 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑂𝑘)) ·o (𝐺‘(𝑂𝑘))) +o 𝑧)), ∅)
4039cantnfvalf 9703 . . . . . 6 𝐻:ω⟶On
4140ffvelcdmi 7103 . . . . 5 ((𝑂𝑋) ∈ ω → (𝐻‘(𝑂𝑋)) ∈ On)
4238, 41syl 17 . . . 4 (𝜑 → (𝐻‘(𝑂𝑋)) ∈ On)
43 oaword1 8589 . . . 4 ((((𝐴o 𝑋) ·o (𝐺𝑋)) ∈ On ∧ (𝐻‘(𝑂𝑋)) ∈ On) → ((𝐴o 𝑋) ·o (𝐺𝑋)) ⊆ (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
4422, 42, 43syl2anc 584 . . 3 (𝜑 → ((𝐴o 𝑋) ·o (𝐺𝑋)) ⊆ (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
453, 1, 2, 24, 6, 39cantnfsuc 9708 . . . . 5 ((𝜑 ∧ (𝑂𝑋) ∈ ω) → (𝐻‘suc (𝑂𝑋)) = (((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) +o (𝐻‘(𝑂𝑋))))
4638, 45mpdan 687 . . . 4 (𝜑 → (𝐻‘suc (𝑂𝑋)) = (((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) +o (𝐻‘(𝑂𝑋))))
47 f1ocnvfv2 7297 . . . . . . . 8 ((𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(𝑂𝑋)) = 𝑋)
4830, 34, 47syl2anc 584 . . . . . . 7 (𝜑 → (𝑂‘(𝑂𝑋)) = 𝑋)
4948oveq2d 7447 . . . . . 6 (𝜑 → (𝐴o (𝑂‘(𝑂𝑋))) = (𝐴o 𝑋))
5048fveq2d 6911 . . . . . 6 (𝜑 → (𝐺‘(𝑂‘(𝑂𝑋))) = (𝐺𝑋))
5149, 50oveq12d 7449 . . . . 5 (𝜑 → ((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) = ((𝐴o 𝑋) ·o (𝐺𝑋)))
5251oveq1d 7446 . . . 4 (𝜑 → (((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) +o (𝐻‘(𝑂𝑋))) = (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
5346, 52eqtrd 2775 . . 3 (𝜑 → (𝐻‘suc (𝑂𝑋)) = (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
5444, 53sseqtrrd 4037 . 2 (𝜑 → ((𝐴o 𝑋) ·o (𝐺𝑋)) ⊆ (𝐻‘suc (𝑂𝑋)))
55 onss 7804 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐵 ⊆ On)
562, 55syl 17 . . . . . . . . . 10 (𝜑𝐵 ⊆ On)
5756sselda 3995 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑥 ∈ On)
5812adantr 480 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑋 ∈ On)
59 onsseleq 6427 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
6057, 58, 59syl2anc 584 . . . . . . . 8 ((𝜑𝑥𝐵) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
61 orcom 870 . . . . . . . 8 ((𝑥𝑋𝑥 = 𝑋) ↔ (𝑥 = 𝑋𝑥𝑋))
6260, 61bitrdi 287 . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥𝑋 ↔ (𝑥 = 𝑋𝑥𝑋)))
6362ifbid 4554 . . . . . 6 ((𝜑𝑥𝐵) → if(𝑥𝑋, (𝐹𝑥), ∅) = if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))
6463mpteq2dva 5248 . . . . 5 (𝜑 → (𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅)) = (𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)))
6564fveq2d 6911 . . . 4 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) = ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))))
663, 1, 2cantnfs 9704 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐵𝐴𝐹 finSupp ∅)))
675, 66mpbid 232 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐵𝐴𝐹 finSupp ∅))
6867simpld 494 . . . . . . . . . 10 (𝜑𝐹:𝐵𝐴)
6968ffvelcdmda 7104 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝐹𝑦) ∈ 𝐴)
7018ne0d 4348 . . . . . . . . . . 11 (𝜑𝐴 ≠ ∅)
71 on0eln0 6442 . . . . . . . . . . . 12 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
721, 71syl 17 . . . . . . . . . . 11 (𝜑 → (∅ ∈ 𝐴𝐴 ≠ ∅))
7370, 72mpbird 257 . . . . . . . . . 10 (𝜑 → ∅ ∈ 𝐴)
7473adantr 480 . . . . . . . . 9 ((𝜑𝑦𝐵) → ∅ ∈ 𝐴)
7569, 74ifcld 4577 . . . . . . . 8 ((𝜑𝑦𝐵) → if(𝑦𝑋, (𝐹𝑦), ∅) ∈ 𝐴)
7675fmpttd 7135 . . . . . . 7 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)):𝐵𝐴)
77 0ex 5313 . . . . . . . . 9 ∅ ∈ V
7877a1i 11 . . . . . . . 8 (𝜑 → ∅ ∈ V)
7967simprd 495 . . . . . . . 8 (𝜑𝐹 finSupp ∅)
8068, 2, 78, 79fsuppmptif 9437 . . . . . . 7 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) finSupp ∅)
813, 1, 2cantnfs 9704 . . . . . . 7 (𝜑 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) ∈ 𝑆 ↔ ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)):𝐵𝐴 ∧ (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) finSupp ∅)))
8276, 80, 81mpbir2and 713 . . . . . 6 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) ∈ 𝑆)
8368, 10ffvelcdmd 7105 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ 𝐴)
84 eldifn 4142 . . . . . . . . 9 (𝑦 ∈ (𝐵𝑋) → ¬ 𝑦𝑋)
8584adantl 481 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐵𝑋)) → ¬ 𝑦𝑋)
8685iffalsed 4542 . . . . . . 7 ((𝜑𝑦 ∈ (𝐵𝑋)) → if(𝑦𝑋, (𝐹𝑦), ∅) = ∅)
8786, 2suppss2 8224 . . . . . 6 (𝜑 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) supp ∅) ⊆ 𝑋)
88 ifor 4585 . . . . . . . 8 if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅))
89 fveq2 6907 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
9089adantl 481 . . . . . . . . . 10 ((𝑥𝐵𝑥 = 𝑋) → (𝐹𝑥) = (𝐹𝑋))
9190ifeq1da 4562 . . . . . . . . 9 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑥), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
92 eleq1w 2822 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦𝑋𝑥𝑋))
93 fveq2 6907 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
9492, 93ifbieq1d 4555 . . . . . . . . . . 11 (𝑦 = 𝑥 → if(𝑦𝑋, (𝐹𝑦), ∅) = if(𝑥𝑋, (𝐹𝑥), ∅))
95 eqid 2735 . . . . . . . . . . 11 (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) = (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))
96 fvex 6920 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
9796, 77ifex 4581 . . . . . . . . . . 11 if(𝑥𝑋, (𝐹𝑥), ∅) ∈ V
9894, 95, 97fvmpt 7016 . . . . . . . . . 10 (𝑥𝐵 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥) = if(𝑥𝑋, (𝐹𝑥), ∅))
9998ifeq2d 4551 . . . . . . . . 9 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑥), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅)))
10091, 99eqtr3d 2777 . . . . . . . 8 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅)))
10188, 100eqtr4id 2794 . . . . . . 7 (𝑥𝐵 → if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅) = if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
102101mpteq2ia 5251 . . . . . 6 (𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)) = (𝑥𝐵 ↦ if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
1033, 1, 2, 82, 10, 83, 87, 102cantnfp1 9719 . . . . 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 495 . . . 4 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))))
10565, 104eqtrd 2775 . . 3 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))))
106 onelon 6411 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐹𝑋) ∈ 𝐴) → (𝐹𝑋) ∈ On)
1071, 83, 106syl2anc 584 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ On)
108 omsuc 8563 . . . . . 6 (((𝐴o 𝑋) ∈ On ∧ (𝐹𝑋) ∈ On) → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)))
10914, 107, 108syl2anc 584 . . . . 5 (𝜑 → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)))
110 eloni 6396 . . . . . . . 8 ((𝐺𝑋) ∈ On → Ord (𝐺𝑋))
11120, 110syl 17 . . . . . . 7 (𝜑 → Ord (𝐺𝑋))
1129simp2d 1142 . . . . . . 7 (𝜑 → (𝐹𝑋) ∈ (𝐺𝑋))
113 ordsucss 7838 . . . . . . 7 (Ord (𝐺𝑋) → ((𝐹𝑋) ∈ (𝐺𝑋) → suc (𝐹𝑋) ⊆ (𝐺𝑋)))
114111, 112, 113sylc 65 . . . . . 6 (𝜑 → suc (𝐹𝑋) ⊆ (𝐺𝑋))
115 onsuc 7831 . . . . . . . 8 ((𝐹𝑋) ∈ On → suc (𝐹𝑋) ∈ On)
116107, 115syl 17 . . . . . . 7 (𝜑 → suc (𝐹𝑋) ∈ On)
117 omwordi 8608 . . . . . . 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 1370 . . . . . 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, 119eqsstrrd 4035 . . . 4 (𝜑 → (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)) ⊆ ((𝐴o 𝑋) ·o (𝐺𝑋)))
1213, 1, 2, 82, 73, 12, 87cantnflt2 9711 . . . . 5 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴o 𝑋))
122 onelon 6411 . . . . . . 7 (((𝐴o 𝑋) ∈ On ∧ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴o 𝑋)) → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On)
12314, 121, 122syl2anc 584 . . . . . 6 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On)
124 omcl 8573 . . . . . . 7 (((𝐴o 𝑋) ∈ On ∧ (𝐹𝑋) ∈ On) → ((𝐴o 𝑋) ·o (𝐹𝑋)) ∈ On)
12514, 107, 124syl2anc 584 . . . . . 6 (𝜑 → ((𝐴o 𝑋) ·o (𝐹𝑋)) ∈ On)
126 oaord 8584 . . . . . 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 1370 . . . . 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 232 . . . 4 (𝜑 → (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)))
129120, 128sseldd 3996 . . 3 (𝜑 → (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ ((𝐴o 𝑋) ·o (𝐺𝑋)))
130105, 129eqeltrd 2839 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) ∈ ((𝐴o 𝑋) ·o (𝐺𝑋)))
13154, 130sseldd 3996 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 206  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  wss 3963  c0 4339  ifcif 4531   cuni 4912   class class class wbr 5148  {copab 5210  cmpt 5231   E cep 5588   We wwe 5640  ccnv 5688  dom cdm 5689  Ord word 6385  Oncon0 6386  suc csuc 6388  wf 6559  1-1-ontowf1o 6562  cfv 6563   Isom wiso 6564  (class class class)co 7431  cmpo 7433  ωcom 7887   supp csupp 8184  seqωcseqom 8486   +o coa 8502   ·o comu 8503  o coe 8504   finSupp cfsupp 9399  OrdIsocoi 9547   CNF ccnf 9699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-cnf 9700
This theorem is referenced by:  cantnflem1  9727
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