Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgval Structured version   Visualization version   GIF version

Theorem efgval 18219
 Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
efgval.w 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
efgval.r = ( ~FG𝐼)
Assertion
Ref Expression
efgval = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))}
Distinct variable groups:   𝑦,𝑟,𝑧,𝑛,𝑥,𝑊   ,𝑟,𝑥,𝑦,𝑧   𝑛,𝐼,𝑟,𝑥,𝑦,𝑧
Allowed substitution hint:   (𝑛)

Proof of Theorem efgval
Dummy variables 𝑖 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.r . 2 = ( ~FG𝐼)
2 vex 3275 . . . . . . . . . . . 12 𝑖 ∈ V
3 2on 7656 . . . . . . . . . . . . 13 2𝑜 ∈ On
43elexi 3285 . . . . . . . . . . . 12 2𝑜 ∈ V
52, 4xpex 7047 . . . . . . . . . . 11 (𝑖 × 2𝑜) ∈ V
6 wrdexg 13390 . . . . . . . . . . 11 ((𝑖 × 2𝑜) ∈ V → Word (𝑖 × 2𝑜) ∈ V)
7 fvi 6337 . . . . . . . . . . 11 (Word (𝑖 × 2𝑜) ∈ V → ( I ‘Word (𝑖 × 2𝑜)) = Word (𝑖 × 2𝑜))
85, 6, 7mp2b 10 . . . . . . . . . 10 ( I ‘Word (𝑖 × 2𝑜)) = Word (𝑖 × 2𝑜)
9 xpeq1 5200 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖 × 2𝑜) = (𝐼 × 2𝑜))
10 wrdeq 13402 . . . . . . . . . . . 12 ((𝑖 × 2𝑜) = (𝐼 × 2𝑜) → Word (𝑖 × 2𝑜) = Word (𝐼 × 2𝑜))
119, 10syl 17 . . . . . . . . . . 11 (𝑖 = 𝐼 → Word (𝑖 × 2𝑜) = Word (𝐼 × 2𝑜))
1211fveq2d 6276 . . . . . . . . . 10 (𝑖 = 𝐼 → ( I ‘Word (𝑖 × 2𝑜)) = ( I ‘Word (𝐼 × 2𝑜)))
138, 12syl5eqr 2740 . . . . . . . . 9 (𝑖 = 𝐼 → Word (𝑖 × 2𝑜) = ( I ‘Word (𝐼 × 2𝑜)))
14 efgval.w . . . . . . . . 9 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
1513, 14syl6eqr 2744 . . . . . . . 8 (𝑖 = 𝐼 → Word (𝑖 × 2𝑜) = 𝑊)
16 ereq2 7838 . . . . . . . 8 (Word (𝑖 × 2𝑜) = 𝑊 → (𝑟 Er Word (𝑖 × 2𝑜) ↔ 𝑟 Er 𝑊))
1715, 16syl 17 . . . . . . 7 (𝑖 = 𝐼 → (𝑟 Er Word (𝑖 × 2𝑜) ↔ 𝑟 Er 𝑊))
18 raleq 3209 . . . . . . . . 9 (𝑖 = 𝐼 → (∀𝑦𝑖𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩) ↔ ∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩)))
1918ralbidv 3056 . . . . . . . 8 (𝑖 = 𝐼 → (∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝑖𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩) ↔ ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩)))
2015, 19raleqbidv 3223 . . . . . . 7 (𝑖 = 𝐼 → (∀𝑥 ∈ Word (𝑖 × 2𝑜)∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝑖𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩) ↔ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩)))
2117, 20anbi12d 749 . . . . . 6 (𝑖 = 𝐼 → ((𝑟 Er Word (𝑖 × 2𝑜) ∧ ∀𝑥 ∈ Word (𝑖 × 2𝑜)∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝑖𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩)) ↔ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))))
2221abbidv 2811 . . . . 5 (𝑖 = 𝐼 → {𝑟 ∣ (𝑟 Er Word (𝑖 × 2𝑜) ∧ ∀𝑥 ∈ Word (𝑖 × 2𝑜)∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝑖𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))})
2322inteqd 4556 . . . 4 (𝑖 = 𝐼 {𝑟 ∣ (𝑟 Er Word (𝑖 × 2𝑜) ∧ ∀𝑥 ∈ Word (𝑖 × 2𝑜)∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝑖𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))})
24 df-efg 18211 . . . 4 ~FG = (𝑖 ∈ V ↦ {𝑟 ∣ (𝑟 Er Word (𝑖 × 2𝑜) ∧ ∀𝑥 ∈ Word (𝑖 × 2𝑜)∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝑖𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))})
2514efglem 18218 . . . . 5 𝑟(𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))
26 intexab 4895 . . . . 5 (∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩)) ↔ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} ∈ V)
2725, 26mpbi 220 . . . 4 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} ∈ V
2823, 24, 27fvmpt 6364 . . 3 (𝐼 ∈ V → ( ~FG𝐼) = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))})
29 fvprc 6266 . . . 4 𝐼 ∈ V → ( ~FG𝐼) = ∅)
30 abn0 4030 . . . . . . . 8 ({𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} ≠ ∅ ↔ ∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩)))
3125, 30mpbir 221 . . . . . . 7 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} ≠ ∅
32 intssuni 4575 . . . . . . 7 ({𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} ≠ ∅ → {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} ⊆ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))})
3331, 32ax-mp 5 . . . . . 6 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} ⊆ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))}
34 erssxp 7853 . . . . . . . . . . . 12 (𝑟 Er 𝑊𝑟 ⊆ (𝑊 × 𝑊))
3514efgrcl 18217 . . . . . . . . . . . . . . . . . 18 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
3635simpld 477 . . . . . . . . . . . . . . . . 17 (𝑥𝑊𝐼 ∈ V)
3736con3i 150 . . . . . . . . . . . . . . . 16 𝐼 ∈ V → ¬ 𝑥𝑊)
3837eq0rdv 4055 . . . . . . . . . . . . . . 15 𝐼 ∈ V → 𝑊 = ∅)
3938xpeq2d 5216 . . . . . . . . . . . . . 14 𝐼 ∈ V → (𝑊 × 𝑊) = (𝑊 × ∅))
40 xp0 5630 . . . . . . . . . . . . . 14 (𝑊 × ∅) = ∅
4139, 40syl6eq 2742 . . . . . . . . . . . . 13 𝐼 ∈ V → (𝑊 × 𝑊) = ∅)
42 ss0b 4049 . . . . . . . . . . . . 13 ((𝑊 × 𝑊) ⊆ ∅ ↔ (𝑊 × 𝑊) = ∅)
4341, 42sylibr 224 . . . . . . . . . . . 12 𝐼 ∈ V → (𝑊 × 𝑊) ⊆ ∅)
4434, 43sylan9ssr 3691 . . . . . . . . . . 11 ((¬ 𝐼 ∈ V ∧ 𝑟 Er 𝑊) → 𝑟 ⊆ ∅)
4544ex 449 . . . . . . . . . 10 𝐼 ∈ V → (𝑟 Er 𝑊𝑟 ⊆ ∅))
4645adantrd 485 . . . . . . . . 9 𝐼 ∈ V → ((𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩)) → 𝑟 ⊆ ∅))
4746alrimiv 1936 . . . . . . . 8 𝐼 ∈ V → ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩)) → 𝑟 ⊆ ∅))
48 sseq1 3700 . . . . . . . . 9 (𝑤 = 𝑟 → (𝑤 ⊆ ∅ ↔ 𝑟 ⊆ ∅))
4948ralab2 3445 . . . . . . . 8 (∀𝑤 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))}𝑤 ⊆ ∅ ↔ ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩)) → 𝑟 ⊆ ∅))
5047, 49sylibr 224 . . . . . . 7 𝐼 ∈ V → ∀𝑤 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))}𝑤 ⊆ ∅)
51 unissb 4545 . . . . . . 7 ( {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} ⊆ ∅ ↔ ∀𝑤 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))}𝑤 ⊆ ∅)
5250, 51sylibr 224 . . . . . 6 𝐼 ∈ V → {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} ⊆ ∅)
5333, 52syl5ss 3688 . . . . 5 𝐼 ∈ V → {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} ⊆ ∅)
54 ss0 4050 . . . . 5 ( {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} ⊆ ∅ → {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} = ∅)
5553, 54syl 17 . . . 4 𝐼 ∈ V → {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))} = ∅)
5629, 55eqtr4d 2729 . . 3 𝐼 ∈ V → ( ~FG𝐼) = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))})
5728, 56pm2.61i 176 . 2 ( ~FG𝐼) = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))}
581, 57eqtri 2714 1 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1562   = wceq 1564  ∃wex 1785   ∈ wcel 2071  {cab 2678   ≠ wne 2864  ∀wral 2982  Vcvv 3272   ∖ cdif 3645   ⊆ wss 3648  ∅c0 3991  ⟨cop 4259  ⟨cotp 4261  ∪ cuni 4512  ∩ cint 4551   class class class wbr 4728   I cid 5095   × cxp 5184  Oncon0 5804  ‘cfv 5969  (class class class)co 6733  1𝑜c1o 7641  2𝑜c2o 7642   Er wer 7827  0cc0 10017  ...cfz 12408  ♯chash 13200  Word cword 13366   splice csplice 13371  ⟨“cs2 13675   ~FG cefg 18208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-rep 4847  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034  ax-cnex 10073  ax-resscn 10074  ax-1cn 10075  ax-icn 10076  ax-addcl 10077  ax-addrcl 10078  ax-mulcl 10079  ax-mulrcl 10080  ax-mulcom 10081  ax-addass 10082  ax-mulass 10083  ax-distr 10084  ax-i2m1 10085  ax-1ne0 10086  ax-1rid 10087  ax-rnegex 10088  ax-rrecex 10089  ax-cnre 10090  ax-pre-lttri 10091  ax-pre-lttrn 10092  ax-pre-ltadd 10093  ax-pre-mulgt0 10094 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-nel 2968  df-ral 2987  df-rex 2988  df-reu 2989  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-pss 3664  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-tp 4258  df-op 4260  df-ot 4262  df-uni 4513  df-int 4552  df-iun 4598  df-br 4729  df-opab 4789  df-mpt 4806  df-tr 4829  df-id 5096  df-eprel 5101  df-po 5107  df-so 5108  df-fr 5145  df-we 5147  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-pred 5761  df-ord 5807  df-on 5808  df-lim 5809  df-suc 5810  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-f1 5974  df-fo 5975  df-f1o 5976  df-fv 5977  df-riota 6694  df-ov 6736  df-oprab 6737  df-mpt2 6738  df-om 7151  df-1st 7253  df-2nd 7254  df-wrecs 7495  df-recs 7556  df-rdg 7594  df-1o 7648  df-2o 7649  df-oadd 7652  df-er 7830  df-map 7944  df-pm 7945  df-en 8041  df-dom 8042  df-sdom 8043  df-fin 8044  df-card 8846  df-pnf 10157  df-mnf 10158  df-xr 10159  df-ltxr 10160  df-le 10161  df-sub 10349  df-neg 10350  df-nn 11102  df-n0 11374  df-z 11459  df-uz 11769  df-fz 12409  df-fzo 12549  df-hash 13201  df-word 13374  df-concat 13376  df-s1 13377  df-substr 13378  df-splice 13379  df-s2 13682  df-efg 18211 This theorem is referenced by:  efger  18220  efgi  18221  efgval2  18226  frgpuplem  18274
 Copyright terms: Public domain W3C validator