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Theorem efgval 19332
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 3437 . . . . . . . . . . . 12 𝑖 ∈ V
3 2on 8320 . . . . . . . . . . . . 13 2o ∈ On
43elexi 3452 . . . . . . . . . . . 12 2o ∈ V
52, 4xpex 7612 . . . . . . . . . . 11 (𝑖 × 2o) ∈ V
6 wrdexg 14236 . . . . . . . . . . 11 ((𝑖 × 2o) ∈ V → Word (𝑖 × 2o) ∈ V)
7 fvi 6853 . . . . . . . . . . 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 5604 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o))
10 wrdeq 14248 . . . . . . . . . . . 12 ((𝑖 × 2o) = (𝐼 × 2o) → Word (𝑖 × 2o) = Word (𝐼 × 2o))
119, 10syl 17 . . . . . . . . . . 11 (𝑖 = 𝐼 → Word (𝑖 × 2o) = Word (𝐼 × 2o))
1211fveq2d 6787 . . . . . . . . . 10 (𝑖 = 𝐼 → ( I ‘Word (𝑖 × 2o)) = ( I ‘Word (𝐼 × 2o)))
138, 12eqtr3id 2793 . . . . . . . . 9 (𝑖 = 𝐼 → Word (𝑖 × 2o) = ( I ‘Word (𝐼 × 2o)))
14 efgval.w . . . . . . . . 9 𝑊 = ( I ‘Word (𝐼 × 2o))
1513, 14eqtr4di 2797 . . . . . . . 8 (𝑖 = 𝐼 → Word (𝑖 × 2o) = 𝑊)
16 ereq2 8515 . . . . . . . 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 3343 . . . . . . . . 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 3113 . . . . . . . 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 3337 . . . . . . 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 631 . . . . . 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 2808 . . . . 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 4885 . . . 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 19324 . . . 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 19331 . . . . 5 𝑟(𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))
26 intexab 5264 . . . . 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 229 . . . 4 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))} ∈ V
2823, 24, 27fvmpt 6884 . . 3 (𝐼 ∈ V → ( ~FG𝐼) = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))})
29 fvprc 6775 . . . 4 𝐼 ∈ V → ( ~FG𝐼) = ∅)
30 abn0 4315 . . . . . . . 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 230 . . . . . . 7 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))} ≠ ∅
32 intssuni 4902 . . . . . . 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 8530 . . . . . . . . . . . 12 (𝑟 Er 𝑊𝑟 ⊆ (𝑊 × 𝑊))
3514efgrcl 19330 . . . . . . . . . . . . . . . . . 18 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
3635simpld 495 . . . . . . . . . . . . . . . . 17 (𝑥𝑊𝐼 ∈ V)
3736con3i 154 . . . . . . . . . . . . . . . 16 𝐼 ∈ V → ¬ 𝑥𝑊)
3837eq0rdv 4339 . . . . . . . . . . . . . . 15 𝐼 ∈ V → 𝑊 = ∅)
3938xpeq2d 5620 . . . . . . . . . . . . . 14 𝐼 ∈ V → (𝑊 × 𝑊) = (𝑊 × ∅))
40 xp0 6066 . . . . . . . . . . . . . 14 (𝑊 × ∅) = ∅
4139, 40eqtrdi 2795 . . . . . . . . . . . . 13 𝐼 ∈ V → (𝑊 × 𝑊) = ∅)
42 ss0b 4332 . . . . . . . . . . . . 13 ((𝑊 × 𝑊) ⊆ ∅ ↔ (𝑊 × 𝑊) = ∅)
4341, 42sylibr 233 . . . . . . . . . . . 12 𝐼 ∈ V → (𝑊 × 𝑊) ⊆ ∅)
4434, 43sylan9ssr 3936 . . . . . . . . . . 11 ((¬ 𝐼 ∈ V ∧ 𝑟 Er 𝑊) → 𝑟 ⊆ ∅)
4544ex 413 . . . . . . . . . 10 𝐼 ∈ V → (𝑟 Er 𝑊𝑟 ⊆ ∅))
4645adantrd 492 . . . . . . . . 9 𝐼 ∈ V → ((𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩)) → 𝑟 ⊆ ∅))
4746alrimiv 1931 . . . . . . . 8 𝐼 ∈ V → ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩)) → 𝑟 ⊆ ∅))
48 sseq1 3947 . . . . . . . . 9 (𝑤 = 𝑟 → (𝑤 ⊆ ∅ ↔ 𝑟 ⊆ ∅))
4948ralab2 3635 . . . . . . . 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 233 . . . . . . 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 4874 . . . . . . 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 233 . . . . . 6 𝐼 ∈ V → {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))} ⊆ ∅)
5333, 52sstrid 3933 . . . . 5 𝐼 ∈ V → {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))} ⊆ ∅)
54 ss0 4333 . . . . 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 2782 . . 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 182 . 2 ( ~FG𝐼) = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))}
581, 57eqtri 2767 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 205  wa 396  wal 1537   = wceq 1539  wex 1782  wcel 2107  {cab 2716  wne 2944  wral 3065  Vcvv 3433  cdif 3885  wss 3888  c0 4257  cop 4568  cotp 4570   cuni 4840   cint 4880   class class class wbr 5075   I cid 5489   × cxp 5588  Oncon0 6270  cfv 6437  (class class class)co 7284  1oc1o 8299  2oc2o 8300   Er wer 8504  0cc0 10880  ...cfz 13248  chash 14053  Word cword 14226   splice csplice 14471  ⟨“cs2 14563   ~FG cefg 19321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-ot 4571  df-uni 4841  df-int 4881  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-om 7722  df-1st 7840  df-2nd 7841  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-2o 8307  df-er 8507  df-map 8626  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-card 9706  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-nn 11983  df-n0 12243  df-z 12329  df-uz 12592  df-fz 13249  df-fzo 13392  df-hash 14054  df-word 14227  df-concat 14283  df-s1 14310  df-substr 14363  df-pfx 14393  df-splice 14472  df-s2 14570  df-efg 19324
This theorem is referenced by:  efger  19333  efgi  19334  efgval2  19339  frgpuplem  19387
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