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

Theorem efgrelexlemb 19532
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 19506 . 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)}
76relopabiv 5776 . . . . . . 7 Rel 𝐿
87a1i 11 . . . . . 6 (⊤ → Rel 𝐿)
9 simpr 485 . . . . . . 7 ((⊤ ∧ 𝑓𝐿𝑔) → 𝑓𝐿𝑔)
10 eqcom 2743 . . . . . . . . . 10 ((𝑎‘0) = (𝑏‘0) ↔ (𝑏‘0) = (𝑎‘0))
11102rexbii 3128 . . . . . . . . 9 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0))
12 rexcom 3273 . . . . . . . . 9 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0) ↔ ∃𝑏 ∈ (𝑆 “ {𝑔})∃𝑎 ∈ (𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
1311, 12bitri 274 . . . . . . . 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 19531 . . . . . . . 8 (𝑓𝐿𝑔 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0))
171, 2, 3, 4, 14, 15, 6efgrelexlema 19531 . . . . . . . 8 (𝑔𝐿𝑓 ↔ ∃𝑏 ∈ (𝑆 “ {𝑔})∃𝑎 ∈ (𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
1813, 16, 173bitr4i 302 . . . . . . 7 (𝑓𝐿𝑔𝑔𝐿𝑓)
199, 18sylib 217 . . . . . 6 ((⊤ ∧ 𝑓𝐿𝑔) → 𝑔𝐿𝑓)
201, 2, 3, 4, 14, 15, 6efgrelexlema 19531 . . . . . . . . 9 (𝑔𝐿 ↔ ∃𝑟 ∈ (𝑆 “ {𝑔})∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0))
21 reeanv 3217 . . . . . . . . . 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 19521 . . . . . . . . . . . . . . . . . . . 20 𝑆:dom 𝑆onto𝑊
23 fofn 6758 . . . . . . . . . . . . . . . . . . . 20 (𝑆:dom 𝑆onto𝑊𝑆 Fn dom 𝑆)
2422, 23ax-mp 5 . . . . . . . . . . . . . . . . . . 19 𝑆 Fn dom 𝑆
25 fniniseg 7010 . . . . . . . . . . . . . . . . . . 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 7010 . . . . . . . . . . . . . . . . . . 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 2762 . . . . . . . . . . . . . . . . . . . 20 (((𝑆𝑟) = 𝑔 ∧ (𝑆𝑏) = 𝑔) → (𝑆𝑟) = (𝑆𝑏))
301, 2, 3, 4, 14, 15efgred 19530 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆 ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑟‘0) = (𝑏‘0))
3130eqcomd 2742 . . . . . . . . . . . . . . . . . . . . 21 ((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆 ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑏‘0) = (𝑟‘0))
32313expa 1118 . . . . . . . . . . . . . . . . . . . 20 (((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆) ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑏‘0) = (𝑟‘0))
3329, 32sylan2 593 . . . . . . . . . . . . . . . . . . 19 (((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆) ∧ ((𝑆𝑟) = 𝑔 ∧ (𝑆𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
3433an4s 658 . . . . . . . . . . . . . . . . . 18 (((𝑟 ∈ dom 𝑆 ∧ (𝑆𝑟) = 𝑔) ∧ (𝑏 ∈ dom 𝑆 ∧ (𝑆𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
3526, 28, 34syl2anb 598 . . . . . . . . . . . . . . . . 17 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → (𝑏‘0) = (𝑟‘0))
36 eqeq2 2748 . . . . . . . . . . . . . . . . 17 ((𝑟‘0) = (𝑠‘0) → ((𝑏‘0) = (𝑟‘0) ↔ (𝑏‘0) = (𝑠‘0)))
3735, 36syl5ibcom 244 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → ((𝑟‘0) = (𝑠‘0) → (𝑏‘0) = (𝑠‘0)))
3837reximdv 3167 . . . . . . . . . . . . . . 15 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑏‘0) = (𝑠‘0)))
39 eqeq1 2740 . . . . . . . . . . . . . . . . 17 ((𝑎‘0) = (𝑏‘0) → ((𝑎‘0) = (𝑠‘0) ↔ (𝑏‘0) = (𝑠‘0)))
4039rexbidv 3175 . . . . . . . . . . . . . . . 16 ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0) ↔ ∃𝑠 ∈ (𝑆 “ {})(𝑏‘0) = (𝑠‘0)))
4140imbi2d 340 . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . . 14 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))))
4342rexlimdva 3152 . . . . . . . . . . . . 13 (𝑟 ∈ (𝑆 “ {𝑔}) → (∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))))
4443impd 411 . . . . . . . . . . . 12 (𝑟 ∈ (𝑆 “ {𝑔}) → ((∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0)))
4544rexlimiv 3145 . . . . . . . . . . 11 (∃𝑟 ∈ (𝑆 “ {𝑔})(∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
4645reximi 3087 . . . . . . . . . 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 234 . . . . . . . . 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 598 . . . . . . . 8 ((𝑓𝐿𝑔𝑔𝐿) → ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
491, 2, 3, 4, 14, 15, 6efgrelexlema 19531 . . . . . . . 8 (𝑓𝐿 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
5048, 49sylibr 233 . . . . . . 7 ((𝑓𝐿𝑔𝑔𝐿) → 𝑓𝐿)
5150adantl 482 . . . . . 6 ((⊤ ∧ (𝑓𝐿𝑔𝑔𝐿)) → 𝑓𝐿)
52 eqid 2736 . . . . . . . . . . . 12 (𝑎‘0) = (𝑎‘0)
53 fveq1 6841 . . . . . . . . . . . . 13 (𝑏 = 𝑎 → (𝑏‘0) = (𝑎‘0))
5453rspceeqv 3595 . . . . . . . . . . . 12 ((𝑎 ∈ (𝑆 “ {𝑓}) ∧ (𝑎‘0) = (𝑎‘0)) → ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
5552, 54mpan2 689 . . . . . . . . . . 11 (𝑎 ∈ (𝑆 “ {𝑓}) → ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
5655pm4.71i 560 . . . . . . . . . 10 (𝑎 ∈ (𝑆 “ {𝑓}) ↔ (𝑎 ∈ (𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)))
57 fniniseg 7010 . . . . . . . . . . 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 276 . . . . . . . . 9 ((𝑎 ∈ (𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆𝑎) = 𝑓))
6059rexbii2 3093 . . . . . . . 8 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓)
611, 2, 3, 4, 14, 15, 6efgrelexlema 19531 . . . . . . . 8 (𝑓𝐿𝑓 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
62 forn 6759 . . . . . . . . . . 11 (𝑆:dom 𝑆onto𝑊 → ran 𝑆 = 𝑊)
6322, 62ax-mp 5 . . . . . . . . . 10 ran 𝑆 = 𝑊
6463eleq2i 2829 . . . . . . . . 9 (𝑓 ∈ ran 𝑆𝑓𝑊)
65 fvelrnb 6903 . . . . . . . . . 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 276 . . . . . . . 8 (𝑓𝑊 ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓)
6860, 61, 673bitr4ri 303 . . . . . . 7 (𝑓𝑊𝑓𝐿𝑓)
6968a1i 11 . . . . . 6 (⊤ → (𝑓𝑊𝑓𝐿𝑓))
708, 19, 51, 69iserd 8674 . . . . 5 (⊤ → 𝐿 Er 𝑊)
7170mptru 1548 . . . 4 𝐿 Er 𝑊
72 simpl 483 . . . . . . . . . . 11 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑎𝑊)
73 foelrn 7056 . . . . . . . . . . 11 ((𝑆:dom 𝑆onto𝑊𝑎𝑊) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆𝑟))
7422, 72, 73sylancr 587 . . . . . . . . . 10 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆𝑟))
75 simprl 769 . . . . . . . . . . 11 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ dom 𝑆)
76 simprr 771 . . . . . . . . . . . 12 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑎 = (𝑆𝑟))
7776eqcomd 2742 . . . . . . . . . . 11 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑆𝑟) = 𝑎)
78 fniniseg 7010 . . . . . . . . . . . 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 583 . . . . . . . . . 10 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ (𝑆 “ {𝑎}))
81 simplr 767 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏 ∈ ran (𝑇𝑎))
8276fveq2d 6846 . . . . . . . . . . . . . . 15 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑇𝑎) = (𝑇‘(𝑆𝑟)))
8382rneqd 5893 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ran (𝑇𝑎) = ran (𝑇‘(𝑆𝑟)))
8481, 83eleqtrd 2840 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏 ∈ ran (𝑇‘(𝑆𝑟)))
851, 2, 3, 4, 14, 15efgsp1 19519 . . . . . . . . . . . . 13 ((𝑟 ∈ dom 𝑆𝑏 ∈ ran (𝑇‘(𝑆𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
8675, 84, 85syl2anc 584 . . . . . . . . . . . 12 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
871, 2, 3, 4, 14, 15efgsdm 19512 . . . . . . . . . . . . . . . 16 (𝑟 ∈ dom 𝑆 ↔ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑟‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑟))(𝑟𝑖) ∈ ran (𝑇‘(𝑟‘(𝑖 − 1)))))
8887simp1bi 1145 . . . . . . . . . . . . . . 15 (𝑟 ∈ dom 𝑆𝑟 ∈ (Word 𝑊 ∖ {∅}))
8988ad2antrl 726 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
9089eldifad 3922 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ Word 𝑊)
911, 2, 3, 4efgtf 19504 . . . . . . . . . . . . . . . . 17 (𝑎𝑊 → ((𝑇𝑎) = (𝑓 ∈ (0...(♯‘𝑎)), 𝑔 ∈ (𝐼 × 2o) ↦ (𝑎 splice ⟨𝑓, 𝑓, ⟨“𝑔(𝑀𝑔)”⟩⟩)) ∧ (𝑇𝑎):((0...(♯‘𝑎)) × (𝐼 × 2o))⟶𝑊))
9291simprd 496 . . . . . . . . . . . . . . . 16 (𝑎𝑊 → (𝑇𝑎):((0...(♯‘𝑎)) × (𝐼 × 2o))⟶𝑊)
9392frnd 6676 . . . . . . . . . . . . . . 15 (𝑎𝑊 → ran (𝑇𝑎) ⊆ 𝑊)
9493sselda 3944 . . . . . . . . . . . . . 14 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑏𝑊)
9594adantr 481 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏𝑊)
961, 2, 3, 4, 14, 15efgsval2 19515 . . . . . . . . . . . . 13 ((𝑟 ∈ Word 𝑊𝑏𝑊 ∧ (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
9790, 95, 86, 96syl3anc 1371 . . . . . . . . . . . 12 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
98 fniniseg 7010 . . . . . . . . . . . . 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 583 . . . . . . . . . . 11 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ (𝑆 “ {𝑏}))
10195s1cld 14491 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ⟨“𝑏”⟩ ∈ Word 𝑊)
102 eldifsn 4747 . . . . . . . . . . . . . . . 16 (𝑟 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝑟 ∈ Word 𝑊𝑟 ≠ ∅))
103 lennncl 14422 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ Word 𝑊𝑟 ≠ ∅) → (♯‘𝑟) ∈ ℕ)
104102, 103sylbi 216 . . . . . . . . . . . . . . 15 (𝑟 ∈ (Word 𝑊 ∖ {∅}) → (♯‘𝑟) ∈ ℕ)
10589, 104syl 17 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (♯‘𝑟) ∈ ℕ)
106 lbfzo0 13612 . . . . . . . . . . . . . 14 (0 ∈ (0..^(♯‘𝑟)) ↔ (♯‘𝑟) ∈ ℕ)
107105, 106sylibr 233 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 0 ∈ (0..^(♯‘𝑟)))
108 ccatval1 14465 . . . . . . . . . . . . 13 ((𝑟 ∈ Word 𝑊 ∧ ⟨“𝑏”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(♯‘𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
10990, 101, 107, 108syl3anc 1371 . . . . . . . . . . . 12 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
110109eqcomd 2742 . . . . . . . . . . 11 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
111 fveq1 6841 . . . . . . . . . . . 12 (𝑠 = (𝑟 ++ ⟨“𝑏”⟩) → (𝑠‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
112111rspceeqv 3595 . . . . . . . . . . 11 (((𝑟 ++ ⟨“𝑏”⟩) ∈ (𝑆 “ {𝑏}) ∧ (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0)) → ∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
113100, 110, 112syl2anc 584 . . . . . . . . . 10 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
11474, 80, 113reximssdv 3169 . . . . . . . . 9 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → ∃𝑟 ∈ (𝑆 “ {𝑎})∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
1151, 2, 3, 4, 14, 15, 6efgrelexlema 19531 . . . . . . . . 9 (𝑎𝐿𝑏 ↔ ∃𝑟 ∈ (𝑆 “ {𝑎})∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
116114, 115sylibr 233 . . . . . . . 8 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑎𝐿𝑏)
117 vex 3449 . . . . . . . . 9 𝑏 ∈ V
118 vex 3449 . . . . . . . . 9 𝑎 ∈ V
119117, 118elec 8692 . . . . . . . 8 (𝑏 ∈ [𝑎]𝐿𝑎𝐿𝑏)
120116, 119sylibr 233 . . . . . . 7 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑏 ∈ [𝑎]𝐿)
121120ex 413 . . . . . 6 (𝑎𝑊 → (𝑏 ∈ ran (𝑇𝑎) → 𝑏 ∈ [𝑎]𝐿))
122121ssrdv 3950 . . . . 5 (𝑎𝑊 → ran (𝑇𝑎) ⊆ [𝑎]𝐿)
123122rgen 3066 . . . 4 𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿
1241fvexi 6856 . . . . . 6 𝑊 ∈ V
125 erex 8672 . . . . . 6 (𝐿 Er 𝑊 → (𝑊 ∈ V → 𝐿 ∈ V))
12671, 124, 125mp2 9 . . . . 5 𝐿 ∈ V
127 ereq1 8655 . . . . . 6 (𝑟 = 𝐿 → (𝑟 Er 𝑊𝐿 Er 𝑊))
128 eceq2 8688 . . . . . . . 8 (𝑟 = 𝐿 → [𝑎]𝑟 = [𝑎]𝐿)
129128sseq2d 3976 . . . . . . 7 (𝑟 = 𝐿 → (ran (𝑇𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇𝑎) ⊆ [𝑎]𝐿))
130129ralbidv 3174 . . . . . 6 (𝑟 = 𝐿 → (∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟 ↔ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿))
131127, 130anbi12d 631 . . . . 5 (𝑟 = 𝐿 → ((𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟) ↔ (𝐿 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿)))
132126, 131elab 3630 . . . 4 (𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} ↔ (𝐿 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿))
13371, 123, 132mpbir2an 709 . . 3 𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)}
134 intss1 4924 . . 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 3978 1 𝐿
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wtru 1542  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  wss 3910  c0 4282  {csn 4586  cop 4592  cotp 4594   cint 4907   ciun 4954   class class class wbr 5105  {copab 5167  cmpt 5188   I cid 5530   × cxp 5631  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  Rel wrel 5638   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  cmpo 7359  1oc1o 8405  2oc2o 8406   Er wer 8645  [cec 8646  0cc0 11051  1c1 11052  cmin 11385  cn 12153  ...cfz 13424  ..^cfzo 13567  chash 14230  Word cword 14402   ++ cconcat 14458  ⟨“cs1 14483   splice csplice 14637  ⟨“cs2 14730   ~FG cefg 19488
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-ec 8650  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-splice 14638  df-s2 14737  df-efg 19491
This theorem is referenced by:  efgrelex  19533
  Copyright terms: Public domain W3C validator