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Theorem efgval2 18065
Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
Hypotheses
Ref Expression
efgval.w 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
efgval.r = ( ~FG𝐼)
efgval2.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
efgval2.t 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
Assertion
Ref Expression
efgval2 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)}
Distinct variable groups:   𝑦,𝑟,𝑧   𝑣,𝑛,𝑤,𝑦,𝑧,𝑟,𝑥   𝑛,𝑀   𝑣,𝑟,𝑤,𝑥,𝑀   𝑇,𝑟,𝑥   𝑛,𝑊,𝑟,𝑣,𝑤   𝑥,𝑦,𝑧,𝑊   ,𝑟,𝑥,𝑦,𝑧   𝑛,𝐼,𝑟,𝑣,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   (𝑤,𝑣,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑀(𝑦,𝑧)

Proof of Theorem efgval2
Dummy variables 𝑎 𝑏 𝑢 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . 3 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
2 efgval.r . . 3 = ( ~FG𝐼)
31, 2efgval 18058 . 2 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑚 ∈ (0...(#‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))}
4 efgval2.m . . . . . . . . . . 11 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
5 efgval2.t . . . . . . . . . . 11 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
61, 2, 4, 5efgtf 18063 . . . . . . . . . 10 (𝑥𝑊 → ((𝑇𝑥) = (𝑚 ∈ (0...(#‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ∧ (𝑇𝑥):((0...(#‘𝑥)) × (𝐼 × 2𝑜))⟶𝑊))
76simpld 475 . . . . . . . . 9 (𝑥𝑊 → (𝑇𝑥) = (𝑚 ∈ (0...(#‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
87rneqd 5318 . . . . . . . 8 (𝑥𝑊 → ran (𝑇𝑥) = ran (𝑚 ∈ (0...(#‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
98sseq1d 3616 . . . . . . 7 (𝑥𝑊 → (ran (𝑇𝑥) ⊆ [𝑥]𝑟 ↔ ran (𝑚 ∈ (0...(#‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ⊆ [𝑥]𝑟))
10 dfss3 3577 . . . . . . . 8 (ran (𝑚 ∈ (0...(#‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ⊆ [𝑥]𝑟 ↔ ∀𝑎 ∈ ran (𝑚 ∈ (0...(#‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟)
11 ovex 6638 . . . . . . . . . . 11 (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ∈ V
1211rgen2w 2920 . . . . . . . . . 10 𝑚 ∈ (0...(#‘𝑥))∀𝑢 ∈ (𝐼 × 2𝑜)(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ∈ V
13 eqid 2621 . . . . . . . . . . 11 (𝑚 ∈ (0...(#‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) = (𝑚 ∈ (0...(#‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))
14 vex 3192 . . . . . . . . . . . . 13 𝑎 ∈ V
15 vex 3192 . . . . . . . . . . . . 13 𝑥 ∈ V
1614, 15elec 7738 . . . . . . . . . . . 12 (𝑎 ∈ [𝑥]𝑟𝑥𝑟𝑎)
17 breq2 4622 . . . . . . . . . . . 12 (𝑎 = (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) → (𝑥𝑟𝑎𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
1816, 17syl5bb 272 . . . . . . . . . . 11 (𝑎 = (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) → (𝑎 ∈ [𝑥]𝑟𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
1913, 18ralrnmpt2 6735 . . . . . . . . . 10 (∀𝑚 ∈ (0...(#‘𝑥))∀𝑢 ∈ (𝐼 × 2𝑜)(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ∈ V → (∀𝑎 ∈ ran (𝑚 ∈ (0...(#‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(#‘𝑥))∀𝑢 ∈ (𝐼 × 2𝑜)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
2012, 19ax-mp 5 . . . . . . . . 9 (∀𝑎 ∈ ran (𝑚 ∈ (0...(#‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(#‘𝑥))∀𝑢 ∈ (𝐼 × 2𝑜)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))
21 id 22 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨𝑎, 𝑏⟩ → 𝑢 = ⟨𝑎, 𝑏⟩)
22 fveq2 6153 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀𝑢) = (𝑀‘⟨𝑎, 𝑏⟩))
23 df-ov 6613 . . . . . . . . . . . . . . . . 17 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
2422, 23syl6eqr 2673 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀𝑢) = (𝑎𝑀𝑏))
2521, 24s2eqd 13552 . . . . . . . . . . . . . . 15 (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨“𝑢(𝑀𝑢)”⟩ = ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩)
2625oteq3d 4389 . . . . . . . . . . . . . 14 (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)
2726oveq2d 6626 . . . . . . . . . . . . 13 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
2827breq2d 4630 . . . . . . . . . . . 12 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)))
2928ralxp 5228 . . . . . . . . . . 11 (∀𝑢 ∈ (𝐼 × 2𝑜)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
30 eqidd 2622 . . . . . . . . . . . . . . . . 17 ((𝑎𝐼𝑏 ∈ 2𝑜) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 𝑏⟩)
314efgmval 18053 . . . . . . . . . . . . . . . . 17 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑎𝑀𝑏) = ⟨𝑎, (1𝑜𝑏)⟩)
3230, 31s2eqd 13552 . . . . . . . . . . . . . . . 16 ((𝑎𝐼𝑏 ∈ 2𝑜) → ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩ = ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)
3332oteq3d 4389 . . . . . . . . . . . . . . 15 ((𝑎𝐼𝑏 ∈ 2𝑜) → ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
3433oveq2d 6626 . . . . . . . . . . . . . 14 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
3534breq2d 4630 . . . . . . . . . . . . 13 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
3635ralbidva 2980 . . . . . . . . . . . 12 (𝑎𝐼 → (∀𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ ∀𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
3736ralbiia 2974 . . . . . . . . . . 11 (∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
3829, 37bitri 264 . . . . . . . . . 10 (∀𝑢 ∈ (𝐼 × 2𝑜)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
3938ralbii 2975 . . . . . . . . 9 (∀𝑚 ∈ (0...(#‘𝑥))∀𝑢 ∈ (𝐼 × 2𝑜)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ ∀𝑚 ∈ (0...(#‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
4020, 39bitri 264 . . . . . . . 8 (∀𝑎 ∈ ran (𝑚 ∈ (0...(#‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(#‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
4110, 40bitri 264 . . . . . . 7 (ran (𝑚 ∈ (0...(#‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ⊆ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(#‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
429, 41syl6bb 276 . . . . . 6 (𝑥𝑊 → (ran (𝑇𝑥) ⊆ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(#‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
4342ralbiia 2974 . . . . 5 (∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟 ↔ ∀𝑥𝑊𝑚 ∈ (0...(#‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
4443anbi2i 729 . . . 4 ((𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟) ↔ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑚 ∈ (0...(#‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
4544abbii 2736 . . 3 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)} = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑚 ∈ (0...(#‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))}
4645inteqi 4449 . 2 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)} = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑚 ∈ (0...(#‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))}
473, 46eqtr4i 2646 1 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  Vcvv 3189  cdif 3556  wss 3559  cop 4159  cotp 4161   cint 4445   class class class wbr 4618  cmpt 4678   I cid 4989   × cxp 5077  ran crn 5080  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612  1𝑜c1o 7505  2𝑜c2o 7506   Er wer 7691  [cec 7692  0cc0 9887  ...cfz 12275  #chash 13064  Word cword 13237   splice csplice 13242  ⟨“cs2 13530   ~FG cefg 18047
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  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-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-ec 7696  df-map 7811  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-n0 11244  df-z 11329  df-uz 11639  df-fz 12276  df-fzo 12414  df-hash 13065  df-word 13245  df-concat 13247  df-s1 13248  df-substr 13249  df-splice 13250  df-s2 13537  df-efg 18050
This theorem is referenced by:  efgi2  18066  efgrelexlemb  18091  efgcpbllemb  18096
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