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

Theorem efgrelexlemb 18872
 Description: If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-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)”⟩⟩)))
efgred.d 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
efgred.s 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
efgrelexlem.1 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}
Assertion
Ref Expression
efgrelexlemb 𝐿
Distinct variable groups:   𝑐,𝑑,𝑖,𝑗   𝑦,𝑧   𝑛,𝑐,𝑡,𝑣,𝑤,𝑦,𝑧,𝑚,𝑥   𝑀,𝑐   𝑖,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑀,𝑗   𝑘,𝑐,𝑇,𝑖,𝑗,𝑚,𝑡,𝑥   𝑊,𝑐   𝑘,𝑑,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧,𝑊,𝑖,𝑗   ,𝑐,𝑑,𝑖,𝑗,𝑚,𝑡,𝑥,𝑦,𝑧   𝑆,𝑐,𝑑,𝑖,𝑗   𝐼,𝑐,𝑖,𝑗,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧   𝐷,𝑐,𝑑,𝑖,𝑗,𝑚,𝑡
Allowed substitution hints:   𝐷(𝑥,𝑦,𝑧,𝑤,𝑣,𝑘,𝑛)   (𝑤,𝑣,𝑘,𝑛)   𝑆(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛,𝑑)   𝐼(𝑘,𝑑)   𝐿(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑖,𝑗,𝑘,𝑚,𝑛,𝑐,𝑑)   𝑀(𝑦,𝑧,𝑘,𝑑)

Proof of Theorem efgrelexlemb
Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑟 𝑠 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)
3 efgval2.m . . 3 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
4 efgval2.t . . 3 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
51, 2, 3, 4efgval2 18846 . 2 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)}
6 efgrelexlem.1 . . . . . . . 8 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}
76relopabi 5659 . . . . . . 7 Rel 𝐿
87a1i 11 . . . . . 6 (⊤ → Rel 𝐿)
9 simpr 488 . . . . . . 7 ((⊤ ∧ 𝑓𝐿𝑔) → 𝑓𝐿𝑔)
10 eqcom 2805 . . . . . . . . . 10 ((𝑎‘0) = (𝑏‘0) ↔ (𝑏‘0) = (𝑎‘0))
11102rexbii 3211 . . . . . . . . 9 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0))
12 rexcom 3308 . . . . . . . . 9 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0) ↔ ∃𝑏 ∈ (𝑆 “ {𝑔})∃𝑎 ∈ (𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
1311, 12bitri 278 . . . . . . . 8 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑏 ∈ (𝑆 “ {𝑔})∃𝑎 ∈ (𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
14 efgred.d . . . . . . . . 9 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
15 efgred.s . . . . . . . . 9 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
161, 2, 3, 4, 14, 15, 6efgrelexlema 18871 . . . . . . . 8 (𝑓𝐿𝑔 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0))
171, 2, 3, 4, 14, 15, 6efgrelexlema 18871 . . . . . . . 8 (𝑔𝐿𝑓 ↔ ∃𝑏 ∈ (𝑆 “ {𝑔})∃𝑎 ∈ (𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
1813, 16, 173bitr4i 306 . . . . . . 7 (𝑓𝐿𝑔𝑔𝐿𝑓)
199, 18sylib 221 . . . . . 6 ((⊤ ∧ 𝑓𝐿𝑔) → 𝑔𝐿𝑓)
201, 2, 3, 4, 14, 15, 6efgrelexlema 18871 . . . . . . . . 9 (𝑔𝐿 ↔ ∃𝑟 ∈ (𝑆 “ {𝑔})∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0))
21 reeanv 3320 . . . . . . . . . 10 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑟 ∈ (𝑆 “ {𝑔})(∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)) ↔ (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑟 ∈ (𝑆 “ {𝑔})∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)))
221, 2, 3, 4, 14, 15efgsfo 18861 . . . . . . . . . . . . . . . . . . . 20 𝑆:dom 𝑆onto𝑊
23 fofn 6568 . . . . . . . . . . . . . . . . . . . 20 (𝑆:dom 𝑆onto𝑊𝑆 Fn dom 𝑆)
2422, 23ax-mp 5 . . . . . . . . . . . . . . . . . . 19 𝑆 Fn dom 𝑆
25 fniniseg 6808 . . . . . . . . . . . . . . . . . . 19 (𝑆 Fn dom 𝑆 → (𝑟 ∈ (𝑆 “ {𝑔}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆𝑟) = 𝑔)))
2624, 25ax-mp 5 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ (𝑆 “ {𝑔}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆𝑟) = 𝑔))
27 fniniseg 6808 . . . . . . . . . . . . . . . . . . 19 (𝑆 Fn dom 𝑆 → (𝑏 ∈ (𝑆 “ {𝑔}) ↔ (𝑏 ∈ dom 𝑆 ∧ (𝑆𝑏) = 𝑔)))
2824, 27ax-mp 5 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ (𝑆 “ {𝑔}) ↔ (𝑏 ∈ dom 𝑆 ∧ (𝑆𝑏) = 𝑔))
29 eqtr3 2820 . . . . . . . . . . . . . . . . . . . 20 (((𝑆𝑟) = 𝑔 ∧ (𝑆𝑏) = 𝑔) → (𝑆𝑟) = (𝑆𝑏))
301, 2, 3, 4, 14, 15efgred 18870 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆 ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑟‘0) = (𝑏‘0))
3130eqcomd 2804 . . . . . . . . . . . . . . . . . . . . 21 ((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆 ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑏‘0) = (𝑟‘0))
32313expa 1115 . . . . . . . . . . . . . . . . . . . 20 (((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆) ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑏‘0) = (𝑟‘0))
3329, 32sylan2 595 . . . . . . . . . . . . . . . . . . 19 (((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆) ∧ ((𝑆𝑟) = 𝑔 ∧ (𝑆𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
3433an4s 659 . . . . . . . . . . . . . . . . . 18 (((𝑟 ∈ dom 𝑆 ∧ (𝑆𝑟) = 𝑔) ∧ (𝑏 ∈ dom 𝑆 ∧ (𝑆𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
3526, 28, 34syl2anb 600 . . . . . . . . . . . . . . . . 17 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → (𝑏‘0) = (𝑟‘0))
36 eqeq2 2810 . . . . . . . . . . . . . . . . 17 ((𝑟‘0) = (𝑠‘0) → ((𝑏‘0) = (𝑟‘0) ↔ (𝑏‘0) = (𝑠‘0)))
3735, 36syl5ibcom 248 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → ((𝑟‘0) = (𝑠‘0) → (𝑏‘0) = (𝑠‘0)))
3837reximdv 3232 . . . . . . . . . . . . . . 15 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑏‘0) = (𝑠‘0)))
39 eqeq1 2802 . . . . . . . . . . . . . . . . 17 ((𝑎‘0) = (𝑏‘0) → ((𝑎‘0) = (𝑠‘0) ↔ (𝑏‘0) = (𝑠‘0)))
4039rexbidv 3256 . . . . . . . . . . . . . . . 16 ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0) ↔ ∃𝑠 ∈ (𝑆 “ {})(𝑏‘0) = (𝑠‘0)))
4140imbi2d 344 . . . . . . . . . . . . . . 15 ((𝑎‘0) = (𝑏‘0) → ((∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0)) ↔ (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑏‘0) = (𝑠‘0))))
4238, 41syl5ibrcom 250 . . . . . . . . . . . . . 14 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))))
4342rexlimdva 3243 . . . . . . . . . . . . 13 (𝑟 ∈ (𝑆 “ {𝑔}) → (∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))))
4443impd 414 . . . . . . . . . . . 12 (𝑟 ∈ (𝑆 “ {𝑔}) → ((∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0)))
4544rexlimiv 3239 . . . . . . . . . . 11 (∃𝑟 ∈ (𝑆 “ {𝑔})(∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
4645reximi 3206 . . . . . . . . . 10 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑟 ∈ (𝑆 “ {𝑔})(∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)) → ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
4721, 46sylbir 238 . . . . . . . . 9 ((∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑟 ∈ (𝑆 “ {𝑔})∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)) → ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
4816, 20, 47syl2anb 600 . . . . . . . 8 ((𝑓𝐿𝑔𝑔𝐿) → ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
491, 2, 3, 4, 14, 15, 6efgrelexlema 18871 . . . . . . . 8 (𝑓𝐿 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
5048, 49sylibr 237 . . . . . . 7 ((𝑓𝐿𝑔𝑔𝐿) → 𝑓𝐿)
5150adantl 485 . . . . . 6 ((⊤ ∧ (𝑓𝐿𝑔𝑔𝐿)) → 𝑓𝐿)
52 eqid 2798 . . . . . . . . . . . 12 (𝑎‘0) = (𝑎‘0)
53 fveq1 6645 . . . . . . . . . . . . 13 (𝑏 = 𝑎 → (𝑏‘0) = (𝑎‘0))
5453rspceeqv 3586 . . . . . . . . . . . 12 ((𝑎 ∈ (𝑆 “ {𝑓}) ∧ (𝑎‘0) = (𝑎‘0)) → ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
5552, 54mpan2 690 . . . . . . . . . . 11 (𝑎 ∈ (𝑆 “ {𝑓}) → ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
5655pm4.71i 563 . . . . . . . . . 10 (𝑎 ∈ (𝑆 “ {𝑓}) ↔ (𝑎 ∈ (𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)))
57 fniniseg 6808 . . . . . . . . . . 11 (𝑆 Fn dom 𝑆 → (𝑎 ∈ (𝑆 “ {𝑓}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆𝑎) = 𝑓)))
5824, 57ax-mp 5 . . . . . . . . . 10 (𝑎 ∈ (𝑆 “ {𝑓}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆𝑎) = 𝑓))
5956, 58bitr3i 280 . . . . . . . . 9 ((𝑎 ∈ (𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆𝑎) = 𝑓))
6059rexbii2 3208 . . . . . . . 8 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓)
611, 2, 3, 4, 14, 15, 6efgrelexlema 18871 . . . . . . . 8 (𝑓𝐿𝑓 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
62 forn 6569 . . . . . . . . . . 11 (𝑆:dom 𝑆onto𝑊 → ran 𝑆 = 𝑊)
6322, 62ax-mp 5 . . . . . . . . . 10 ran 𝑆 = 𝑊
6463eleq2i 2881 . . . . . . . . 9 (𝑓 ∈ ran 𝑆𝑓𝑊)
65 fvelrnb 6702 . . . . . . . . . 10 (𝑆 Fn dom 𝑆 → (𝑓 ∈ ran 𝑆 ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓))
6624, 65ax-mp 5 . . . . . . . . 9 (𝑓 ∈ ran 𝑆 ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓)
6764, 66bitr3i 280 . . . . . . . 8 (𝑓𝑊 ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓)
6860, 61, 673bitr4ri 307 . . . . . . 7 (𝑓𝑊𝑓𝐿𝑓)
6968a1i 11 . . . . . 6 (⊤ → (𝑓𝑊𝑓𝐿𝑓))
708, 19, 51, 69iserd 8301 . . . . 5 (⊤ → 𝐿 Er 𝑊)
7170mptru 1545 . . . 4 𝐿 Er 𝑊
72 simpl 486 . . . . . . . . . . 11 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑎𝑊)
73 foelrn 6850 . . . . . . . . . . 11 ((𝑆:dom 𝑆onto𝑊𝑎𝑊) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆𝑟))
7422, 72, 73sylancr 590 . . . . . . . . . 10 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆𝑟))
75 simprl 770 . . . . . . . . . . 11 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ dom 𝑆)
76 simprr 772 . . . . . . . . . . . 12 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑎 = (𝑆𝑟))
7776eqcomd 2804 . . . . . . . . . . 11 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑆𝑟) = 𝑎)
78 fniniseg 6808 . . . . . . . . . . . 12 (𝑆 Fn dom 𝑆 → (𝑟 ∈ (𝑆 “ {𝑎}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆𝑟) = 𝑎)))
7924, 78ax-mp 5 . . . . . . . . . . 11 (𝑟 ∈ (𝑆 “ {𝑎}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆𝑟) = 𝑎))
8075, 77, 79sylanbrc 586 . . . . . . . . . 10 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ (𝑆 “ {𝑎}))
81 simplr 768 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏 ∈ ran (𝑇𝑎))
8276fveq2d 6650 . . . . . . . . . . . . . . 15 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑇𝑎) = (𝑇‘(𝑆𝑟)))
8382rneqd 5773 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ran (𝑇𝑎) = ran (𝑇‘(𝑆𝑟)))
8481, 83eleqtrd 2892 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏 ∈ ran (𝑇‘(𝑆𝑟)))
851, 2, 3, 4, 14, 15efgsp1 18859 . . . . . . . . . . . . 13 ((𝑟 ∈ dom 𝑆𝑏 ∈ ran (𝑇‘(𝑆𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
8675, 84, 85syl2anc 587 . . . . . . . . . . . 12 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
871, 2, 3, 4, 14, 15efgsdm 18852 . . . . . . . . . . . . . . . 16 (𝑟 ∈ dom 𝑆 ↔ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑟‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑟))(𝑟𝑖) ∈ ran (𝑇‘(𝑟‘(𝑖 − 1)))))
8887simp1bi 1142 . . . . . . . . . . . . . . 15 (𝑟 ∈ dom 𝑆𝑟 ∈ (Word 𝑊 ∖ {∅}))
8988ad2antrl 727 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
9089eldifad 3893 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ Word 𝑊)
911, 2, 3, 4efgtf 18844 . . . . . . . . . . . . . . . . 17 (𝑎𝑊 → ((𝑇𝑎) = (𝑓 ∈ (0...(♯‘𝑎)), 𝑔 ∈ (𝐼 × 2o) ↦ (𝑎 splice ⟨𝑓, 𝑓, ⟨“𝑔(𝑀𝑔)”⟩⟩)) ∧ (𝑇𝑎):((0...(♯‘𝑎)) × (𝐼 × 2o))⟶𝑊))
9291simprd 499 . . . . . . . . . . . . . . . 16 (𝑎𝑊 → (𝑇𝑎):((0...(♯‘𝑎)) × (𝐼 × 2o))⟶𝑊)
9392frnd 6495 . . . . . . . . . . . . . . 15 (𝑎𝑊 → ran (𝑇𝑎) ⊆ 𝑊)
9493sselda 3915 . . . . . . . . . . . . . 14 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑏𝑊)
9594adantr 484 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏𝑊)
961, 2, 3, 4, 14, 15efgsval2 18855 . . . . . . . . . . . . 13 ((𝑟 ∈ Word 𝑊𝑏𝑊 ∧ (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
9790, 95, 86, 96syl3anc 1368 . . . . . . . . . . . 12 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
98 fniniseg 6808 . . . . . . . . . . . . 13 (𝑆 Fn dom 𝑆 → ((𝑟 ++ ⟨“𝑏”⟩) ∈ (𝑆 “ {𝑏}) ↔ ((𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆 ∧ (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)))
9924, 98ax-mp 5 . . . . . . . . . . . 12 ((𝑟 ++ ⟨“𝑏”⟩) ∈ (𝑆 “ {𝑏}) ↔ ((𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆 ∧ (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏))
10086, 97, 99sylanbrc 586 . . . . . . . . . . 11 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ (𝑆 “ {𝑏}))
10195s1cld 13951 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ⟨“𝑏”⟩ ∈ Word 𝑊)
102 eldifsn 4680 . . . . . . . . . . . . . . . 16 (𝑟 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝑟 ∈ Word 𝑊𝑟 ≠ ∅))
103 lennncl 13880 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ Word 𝑊𝑟 ≠ ∅) → (♯‘𝑟) ∈ ℕ)
104102, 103sylbi 220 . . . . . . . . . . . . . . 15 (𝑟 ∈ (Word 𝑊 ∖ {∅}) → (♯‘𝑟) ∈ ℕ)
10589, 104syl 17 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (♯‘𝑟) ∈ ℕ)
106 lbfzo0 13075 . . . . . . . . . . . . . 14 (0 ∈ (0..^(♯‘𝑟)) ↔ (♯‘𝑟) ∈ ℕ)
107105, 106sylibr 237 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 0 ∈ (0..^(♯‘𝑟)))
108 ccatval1 13924 . . . . . . . . . . . . 13 ((𝑟 ∈ Word 𝑊 ∧ ⟨“𝑏”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(♯‘𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
10990, 101, 107, 108syl3anc 1368 . . . . . . . . . . . 12 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
110109eqcomd 2804 . . . . . . . . . . 11 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
111 fveq1 6645 . . . . . . . . . . . 12 (𝑠 = (𝑟 ++ ⟨“𝑏”⟩) → (𝑠‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
112111rspceeqv 3586 . . . . . . . . . . 11 (((𝑟 ++ ⟨“𝑏”⟩) ∈ (𝑆 “ {𝑏}) ∧ (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0)) → ∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
113100, 110, 112syl2anc 587 . . . . . . . . . 10 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
11474, 80, 113reximssdv 3235 . . . . . . . . 9 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → ∃𝑟 ∈ (𝑆 “ {𝑎})∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
1151, 2, 3, 4, 14, 15, 6efgrelexlema 18871 . . . . . . . . 9 (𝑎𝐿𝑏 ↔ ∃𝑟 ∈ (𝑆 “ {𝑎})∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
116114, 115sylibr 237 . . . . . . . 8 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑎𝐿𝑏)
117 vex 3444 . . . . . . . . 9 𝑏 ∈ V
118 vex 3444 . . . . . . . . 9 𝑎 ∈ V
119117, 118elec 8319 . . . . . . . 8 (𝑏 ∈ [𝑎]𝐿𝑎𝐿𝑏)
120116, 119sylibr 237 . . . . . . 7 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑏 ∈ [𝑎]𝐿)
121120ex 416 . . . . . 6 (𝑎𝑊 → (𝑏 ∈ ran (𝑇𝑎) → 𝑏 ∈ [𝑎]𝐿))
122121ssrdv 3921 . . . . 5 (𝑎𝑊 → ran (𝑇𝑎) ⊆ [𝑎]𝐿)
123122rgen 3116 . . . 4 𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿
1241fvexi 6660 . . . . . 6 𝑊 ∈ V
125 erex 8299 . . . . . 6 (𝐿 Er 𝑊 → (𝑊 ∈ V → 𝐿 ∈ V))
12671, 124, 125mp2 9 . . . . 5 𝐿 ∈ V
127 ereq1 8282 . . . . . 6 (𝑟 = 𝐿 → (𝑟 Er 𝑊𝐿 Er 𝑊))
128 eceq2 8315 . . . . . . . 8 (𝑟 = 𝐿 → [𝑎]𝑟 = [𝑎]𝐿)
129128sseq2d 3947 . . . . . . 7 (𝑟 = 𝐿 → (ran (𝑇𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇𝑎) ⊆ [𝑎]𝐿))
130129ralbidv 3162 . . . . . 6 (𝑟 = 𝐿 → (∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟 ↔ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿))
131127, 130anbi12d 633 . . . . 5 (𝑟 = 𝐿 → ((𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟) ↔ (𝐿 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿)))
132126, 131elab 3615 . . . 4 (𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} ↔ (𝐿 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿))
13371, 123, 132mpbir2an 710 . . 3 𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)}
134 intss1 4854 . . 3 (𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} → {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} ⊆ 𝐿)
135133, 134ax-mp 5 . 2 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} ⊆ 𝐿
1365, 135eqsstri 3949 1 𝐿
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  {csn 4525  ⟨cop 4531  ⟨cotp 4533  ∩ cint 4839  ∪ ciun 4882   class class class wbr 5031  {copab 5093   ↦ cmpt 5111   I cid 5425   × cxp 5518  ◡ccnv 5519  dom cdm 5520  ran crn 5521   “ cima 5523  Rel wrel 5525   Fn wfn 6320  ⟶wf 6321  –onto→wfo 6323  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  1oc1o 8081  2oc2o 8082   Er wer 8272  [cec 8273  0cc0 10529  1c1 10530   − cmin 10862  ℕcn 11628  ...cfz 12888  ..^cfzo 13031  ♯chash 13689  Word cword 13860   ++ cconcat 13916  ⟨“cs1 13943   splice csplice 14105  ⟨“cs2 14197   ~FG cefg 18828 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-ot 4534  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-2o 8089  df-oadd 8092  df-er 8275  df-ec 8277  df-map 8394  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-2 11691  df-n0 11889  df-xnn0 11959  df-z 11973  df-uz 12235  df-rp 12381  df-fz 12889  df-fzo 13032  df-hash 13690  df-word 13861  df-concat 13917  df-s1 13944  df-substr 13997  df-pfx 14027  df-splice 14106  df-s2 14204  df-efg 18831 This theorem is referenced by:  efgrelex  18873
 Copyright terms: Public domain W3C validator