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Theorem efgval 18961
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 3402 . . . . . . . . . . . 12 𝑖 ∈ V
3 2on 8139 . . . . . . . . . . . . 13 2o ∈ On
43elexi 3417 . . . . . . . . . . . 12 2o ∈ V
52, 4xpex 7494 . . . . . . . . . . 11 (𝑖 × 2o) ∈ V
6 wrdexg 13965 . . . . . . . . . . 11 ((𝑖 × 2o) ∈ V → Word (𝑖 × 2o) ∈ V)
7 fvi 6744 . . . . . . . . . . 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 5539 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o))
10 wrdeq 13977 . . . . . . . . . . . 12 ((𝑖 × 2o) = (𝐼 × 2o) → Word (𝑖 × 2o) = Word (𝐼 × 2o))
119, 10syl 17 . . . . . . . . . . 11 (𝑖 = 𝐼 → Word (𝑖 × 2o) = Word (𝐼 × 2o))
1211fveq2d 6678 . . . . . . . . . 10 (𝑖 = 𝐼 → ( I ‘Word (𝑖 × 2o)) = ( I ‘Word (𝐼 × 2o)))
138, 12eqtr3id 2787 . . . . . . . . 9 (𝑖 = 𝐼 → Word (𝑖 × 2o) = ( I ‘Word (𝐼 × 2o)))
14 efgval.w . . . . . . . . 9 𝑊 = ( I ‘Word (𝐼 × 2o))
1513, 14eqtr4di 2791 . . . . . . . 8 (𝑖 = 𝐼 → Word (𝑖 × 2o) = 𝑊)
16 ereq2 8328 . . . . . . . 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 3310 . . . . . . . . 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 3109 . . . . . . . 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 3304 . . . . . . 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 634 . . . . . 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 2802 . . . . 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 4841 . . . 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 18953 . . . 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 18960 . . . . 5 𝑟(𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))
26 intexab 5207 . . . . 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 233 . . . 4 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))} ∈ V
2823, 24, 27fvmpt 6775 . . 3 (𝐼 ∈ V → ( ~FG𝐼) = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))})
29 fvprc 6666 . . . 4 𝐼 ∈ V → ( ~FG𝐼) = ∅)
30 abn0 4269 . . . . . . . 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 234 . . . . . . 7 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))} ≠ ∅
32 intssuni 4858 . . . . . . 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 8343 . . . . . . . . . . . 12 (𝑟 Er 𝑊𝑟 ⊆ (𝑊 × 𝑊))
3514efgrcl 18959 . . . . . . . . . . . . . . . . . 18 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
3635simpld 498 . . . . . . . . . . . . . . . . 17 (𝑥𝑊𝐼 ∈ V)
3736con3i 157 . . . . . . . . . . . . . . . 16 𝐼 ∈ V → ¬ 𝑥𝑊)
3837eq0rdv 4293 . . . . . . . . . . . . . . 15 𝐼 ∈ V → 𝑊 = ∅)
3938xpeq2d 5555 . . . . . . . . . . . . . 14 𝐼 ∈ V → (𝑊 × 𝑊) = (𝑊 × ∅))
40 xp0 5990 . . . . . . . . . . . . . 14 (𝑊 × ∅) = ∅
4139, 40eqtrdi 2789 . . . . . . . . . . . . 13 𝐼 ∈ V → (𝑊 × 𝑊) = ∅)
42 ss0b 4286 . . . . . . . . . . . . 13 ((𝑊 × 𝑊) ⊆ ∅ ↔ (𝑊 × 𝑊) = ∅)
4341, 42sylibr 237 . . . . . . . . . . . 12 𝐼 ∈ V → (𝑊 × 𝑊) ⊆ ∅)
4434, 43sylan9ssr 3891 . . . . . . . . . . 11 ((¬ 𝐼 ∈ V ∧ 𝑟 Er 𝑊) → 𝑟 ⊆ ∅)
4544ex 416 . . . . . . . . . 10 𝐼 ∈ V → (𝑟 Er 𝑊𝑟 ⊆ ∅))
4645adantrd 495 . . . . . . . . 9 𝐼 ∈ V → ((𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩)) → 𝑟 ⊆ ∅))
4746alrimiv 1934 . . . . . . . 8 𝐼 ∈ V → ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩)) → 𝑟 ⊆ ∅))
48 sseq1 3902 . . . . . . . . 9 (𝑤 = 𝑟 → (𝑤 ⊆ ∅ ↔ 𝑟 ⊆ ∅))
4948ralab2 3596 . . . . . . . 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 237 . . . . . . 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 4830 . . . . . . 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 237 . . . . . 6 𝐼 ∈ V → {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))} ⊆ ∅)
5333, 52sstrid 3888 . . . . 5 𝐼 ∈ V → {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))} ⊆ ∅)
54 ss0 4287 . . . . 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 2776 . . 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 185 . 2 ( ~FG𝐼) = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))}
581, 57eqtri 2761 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 209  wa 399  wal 1540   = wceq 1542  wex 1786  wcel 2114  {cab 2716  wne 2934  wral 3053  Vcvv 3398  cdif 3840  wss 3843  c0 4211  cop 4522  cotp 4524   cuni 4796   cint 4836   class class class wbr 5030   I cid 5428   × cxp 5523  Oncon0 6172  cfv 6339  (class class class)co 7170  1oc1o 8124  2oc2o 8125   Er wer 8317  0cc0 10615  ...cfz 12981  chash 13782  Word cword 13955   splice csplice 14200  ⟨“cs2 14292   ~FG cefg 18950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-ot 4525  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-2o 8132  df-er 8320  df-map 8439  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-n0 11977  df-z 12063  df-uz 12325  df-fz 12982  df-fzo 13125  df-hash 13783  df-word 13956  df-concat 14012  df-s1 14039  df-substr 14092  df-pfx 14122  df-splice 14201  df-s2 14299  df-efg 18953
This theorem is referenced by:  efger  18962  efgi  18963  efgval2  18968  frgpuplem  19016
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