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Theorem efgrelexlemb 19783
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 19757 . 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 5833 . . . . . . 7 Rel 𝐿
87a1i 11 . . . . . 6 (⊤ → Rel 𝐿)
9 simpr 484 . . . . . . 7 ((⊤ ∧ 𝑓𝐿𝑔) → 𝑓𝐿𝑔)
10 eqcom 2742 . . . . . . . . . 10 ((𝑎‘0) = (𝑏‘0) ↔ (𝑏‘0) = (𝑎‘0))
11102rexbii 3127 . . . . . . . . 9 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0))
12 rexcom 3288 . . . . . . . . 9 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0) ↔ ∃𝑏 ∈ (𝑆 “ {𝑔})∃𝑎 ∈ (𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
1311, 12bitri 275 . . . . . . . 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 19782 . . . . . . . 8 (𝑓𝐿𝑔 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0))
171, 2, 3, 4, 14, 15, 6efgrelexlema 19782 . . . . . . . 8 (𝑔𝐿𝑓 ↔ ∃𝑏 ∈ (𝑆 “ {𝑔})∃𝑎 ∈ (𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
1813, 16, 173bitr4i 303 . . . . . . 7 (𝑓𝐿𝑔𝑔𝐿𝑓)
199, 18sylib 218 . . . . . 6 ((⊤ ∧ 𝑓𝐿𝑔) → 𝑔𝐿𝑓)
201, 2, 3, 4, 14, 15, 6efgrelexlema 19782 . . . . . . . . 9 (𝑔𝐿 ↔ ∃𝑟 ∈ (𝑆 “ {𝑔})∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0))
21 reeanv 3227 . . . . . . . . . 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 19772 . . . . . . . . . . . . . . . . . . . 20 𝑆:dom 𝑆onto𝑊
23 fofn 6823 . . . . . . . . . . . . . . . . . . . 20 (𝑆:dom 𝑆onto𝑊𝑆 Fn dom 𝑆)
2422, 23ax-mp 5 . . . . . . . . . . . . . . . . . . 19 𝑆 Fn dom 𝑆
25 fniniseg 7080 . . . . . . . . . . . . . . . . . . 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 7080 . . . . . . . . . . . . . . . . . . 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 2761 . . . . . . . . . . . . . . . . . . . 20 (((𝑆𝑟) = 𝑔 ∧ (𝑆𝑏) = 𝑔) → (𝑆𝑟) = (𝑆𝑏))
301, 2, 3, 4, 14, 15efgred 19781 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆 ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑟‘0) = (𝑏‘0))
3130eqcomd 2741 . . . . . . . . . . . . . . . . . . . . 21 ((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆 ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑏‘0) = (𝑟‘0))
32313expa 1117 . . . . . . . . . . . . . . . . . . . 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 660 . . . . . . . . . . . . . . . . . 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 2747 . . . . . . . . . . . . . . . . 17 ((𝑟‘0) = (𝑠‘0) → ((𝑏‘0) = (𝑟‘0) ↔ (𝑏‘0) = (𝑠‘0)))
3735, 36syl5ibcom 245 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → ((𝑟‘0) = (𝑠‘0) → (𝑏‘0) = (𝑠‘0)))
3837reximdv 3168 . . . . . . . . . . . . . . 15 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑏‘0) = (𝑠‘0)))
39 eqeq1 2739 . . . . . . . . . . . . . . . . 17 ((𝑎‘0) = (𝑏‘0) → ((𝑎‘0) = (𝑠‘0) ↔ (𝑏‘0) = (𝑠‘0)))
4039rexbidv 3177 . . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . 14 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))))
4342rexlimdva 3153 . . . . . . . . . . . . 13 (𝑟 ∈ (𝑆 “ {𝑔}) → (∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))))
4443impd 410 . . . . . . . . . . . 12 (𝑟 ∈ (𝑆 “ {𝑔}) → ((∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0)))
4544rexlimiv 3146 . . . . . . . . . . 11 (∃𝑟 ∈ (𝑆 “ {𝑔})(∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
4645reximi 3082 . . . . . . . . . 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 235 . . . . . . . . 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 19782 . . . . . . . 8 (𝑓𝐿 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
5048, 49sylibr 234 . . . . . . 7 ((𝑓𝐿𝑔𝑔𝐿) → 𝑓𝐿)
5150adantl 481 . . . . . 6 ((⊤ ∧ (𝑓𝐿𝑔𝑔𝐿)) → 𝑓𝐿)
52 eqid 2735 . . . . . . . . . . . 12 (𝑎‘0) = (𝑎‘0)
53 fveq1 6906 . . . . . . . . . . . . 13 (𝑏 = 𝑎 → (𝑏‘0) = (𝑎‘0))
5453rspceeqv 3645 . . . . . . . . . . . 12 ((𝑎 ∈ (𝑆 “ {𝑓}) ∧ (𝑎‘0) = (𝑎‘0)) → ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
5552, 54mpan2 691 . . . . . . . . . . 11 (𝑎 ∈ (𝑆 “ {𝑓}) → ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
5655pm4.71i 559 . . . . . . . . . 10 (𝑎 ∈ (𝑆 “ {𝑓}) ↔ (𝑎 ∈ (𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)))
57 fniniseg 7080 . . . . . . . . . . 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 277 . . . . . . . . 9 ((𝑎 ∈ (𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆𝑎) = 𝑓))
6059rexbii2 3088 . . . . . . . 8 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓)
611, 2, 3, 4, 14, 15, 6efgrelexlema 19782 . . . . . . . 8 (𝑓𝐿𝑓 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
62 forn 6824 . . . . . . . . . . 11 (𝑆:dom 𝑆onto𝑊 → ran 𝑆 = 𝑊)
6322, 62ax-mp 5 . . . . . . . . . 10 ran 𝑆 = 𝑊
6463eleq2i 2831 . . . . . . . . 9 (𝑓 ∈ ran 𝑆𝑓𝑊)
65 fvelrnb 6969 . . . . . . . . . 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 277 . . . . . . . 8 (𝑓𝑊 ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓)
6860, 61, 673bitr4ri 304 . . . . . . 7 (𝑓𝑊𝑓𝐿𝑓)
6968a1i 11 . . . . . 6 (⊤ → (𝑓𝑊𝑓𝐿𝑓))
708, 19, 51, 69iserd 8770 . . . . 5 (⊤ → 𝐿 Er 𝑊)
7170mptru 1544 . . . 4 𝐿 Er 𝑊
72 simpl 482 . . . . . . . . . . 11 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑎𝑊)
73 foelrn 7127 . . . . . . . . . . 11 ((𝑆:dom 𝑆onto𝑊𝑎𝑊) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆𝑟))
7422, 72, 73sylancr 587 . . . . . . . . . 10 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆𝑟))
75 simprl 771 . . . . . . . . . . 11 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ dom 𝑆)
76 simprr 773 . . . . . . . . . . . 12 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑎 = (𝑆𝑟))
7776eqcomd 2741 . . . . . . . . . . 11 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑆𝑟) = 𝑎)
78 fniniseg 7080 . . . . . . . . . . . 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 769 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏 ∈ ran (𝑇𝑎))
8276fveq2d 6911 . . . . . . . . . . . . . . 15 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑇𝑎) = (𝑇‘(𝑆𝑟)))
8382rneqd 5952 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ran (𝑇𝑎) = ran (𝑇‘(𝑆𝑟)))
8481, 83eleqtrd 2841 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏 ∈ ran (𝑇‘(𝑆𝑟)))
851, 2, 3, 4, 14, 15efgsp1 19770 . . . . . . . . . . . . 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 19763 . . . . . . . . . . . . . . . 16 (𝑟 ∈ dom 𝑆 ↔ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑟‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑟))(𝑟𝑖) ∈ ran (𝑇‘(𝑟‘(𝑖 − 1)))))
8887simp1bi 1144 . . . . . . . . . . . . . . 15 (𝑟 ∈ dom 𝑆𝑟 ∈ (Word 𝑊 ∖ {∅}))
8988ad2antrl 728 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
9089eldifad 3975 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ Word 𝑊)
911, 2, 3, 4efgtf 19755 . . . . . . . . . . . . . . . . 17 (𝑎𝑊 → ((𝑇𝑎) = (𝑓 ∈ (0...(♯‘𝑎)), 𝑔 ∈ (𝐼 × 2o) ↦ (𝑎 splice ⟨𝑓, 𝑓, ⟨“𝑔(𝑀𝑔)”⟩⟩)) ∧ (𝑇𝑎):((0...(♯‘𝑎)) × (𝐼 × 2o))⟶𝑊))
9291simprd 495 . . . . . . . . . . . . . . . 16 (𝑎𝑊 → (𝑇𝑎):((0...(♯‘𝑎)) × (𝐼 × 2o))⟶𝑊)
9392frnd 6745 . . . . . . . . . . . . . . 15 (𝑎𝑊 → ran (𝑇𝑎) ⊆ 𝑊)
9493sselda 3995 . . . . . . . . . . . . . 14 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑏𝑊)
9594adantr 480 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏𝑊)
961, 2, 3, 4, 14, 15efgsval2 19766 . . . . . . . . . . . . 13 ((𝑟 ∈ Word 𝑊𝑏𝑊 ∧ (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
9790, 95, 86, 96syl3anc 1370 . . . . . . . . . . . 12 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
98 fniniseg 7080 . . . . . . . . . . . . 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 14638 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ⟨“𝑏”⟩ ∈ Word 𝑊)
102 eldifsn 4791 . . . . . . . . . . . . . . . 16 (𝑟 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝑟 ∈ Word 𝑊𝑟 ≠ ∅))
103 lennncl 14569 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ Word 𝑊𝑟 ≠ ∅) → (♯‘𝑟) ∈ ℕ)
104102, 103sylbi 217 . . . . . . . . . . . . . . 15 (𝑟 ∈ (Word 𝑊 ∖ {∅}) → (♯‘𝑟) ∈ ℕ)
10589, 104syl 17 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (♯‘𝑟) ∈ ℕ)
106 lbfzo0 13736 . . . . . . . . . . . . . 14 (0 ∈ (0..^(♯‘𝑟)) ↔ (♯‘𝑟) ∈ ℕ)
107105, 106sylibr 234 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 0 ∈ (0..^(♯‘𝑟)))
108 ccatval1 14612 . . . . . . . . . . . . 13 ((𝑟 ∈ Word 𝑊 ∧ ⟨“𝑏”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(♯‘𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
10990, 101, 107, 108syl3anc 1370 . . . . . . . . . . . 12 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
110109eqcomd 2741 . . . . . . . . . . 11 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
111 fveq1 6906 . . . . . . . . . . . 12 (𝑠 = (𝑟 ++ ⟨“𝑏”⟩) → (𝑠‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
112111rspceeqv 3645 . . . . . . . . . . 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 3171 . . . . . . . . 9 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → ∃𝑟 ∈ (𝑆 “ {𝑎})∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
1151, 2, 3, 4, 14, 15, 6efgrelexlema 19782 . . . . . . . . 9 (𝑎𝐿𝑏 ↔ ∃𝑟 ∈ (𝑆 “ {𝑎})∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
116114, 115sylibr 234 . . . . . . . 8 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑎𝐿𝑏)
117 vex 3482 . . . . . . . . 9 𝑏 ∈ V
118 vex 3482 . . . . . . . . 9 𝑎 ∈ V
119117, 118elec 8790 . . . . . . . 8 (𝑏 ∈ [𝑎]𝐿𝑎𝐿𝑏)
120116, 119sylibr 234 . . . . . . 7 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑏 ∈ [𝑎]𝐿)
121120ex 412 . . . . . 6 (𝑎𝑊 → (𝑏 ∈ ran (𝑇𝑎) → 𝑏 ∈ [𝑎]𝐿))
122121ssrdv 4001 . . . . 5 (𝑎𝑊 → ran (𝑇𝑎) ⊆ [𝑎]𝐿)
123122rgen 3061 . . . 4 𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿
1241fvexi 6921 . . . . . 6 𝑊 ∈ V
125 erex 8768 . . . . . 6 (𝐿 Er 𝑊 → (𝑊 ∈ V → 𝐿 ∈ V))
12671, 124, 125mp2 9 . . . . 5 𝐿 ∈ V
127 ereq1 8751 . . . . . 6 (𝑟 = 𝐿 → (𝑟 Er 𝑊𝐿 Er 𝑊))
128 eceq2 8785 . . . . . . . 8 (𝑟 = 𝐿 → [𝑎]𝑟 = [𝑎]𝐿)
129128sseq2d 4028 . . . . . . 7 (𝑟 = 𝐿 → (ran (𝑇𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇𝑎) ⊆ [𝑎]𝐿))
130129ralbidv 3176 . . . . . 6 (𝑟 = 𝐿 → (∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟 ↔ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿))
131127, 130anbi12d 632 . . . . 5 (𝑟 = 𝐿 → ((𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟) ↔ (𝐿 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿)))
132126, 131elab 3681 . . . 4 (𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} ↔ (𝐿 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿))
13371, 123, 132mpbir2an 711 . . 3 𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)}
134 intss1 4968 . . 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 4030 1 𝐿
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wtru 1538  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  wss 3963  c0 4339  {csn 4631  cop 4637  cotp 4639   cint 4951   ciun 4996   class class class wbr 5148  {copab 5210  cmpt 5231   I cid 5582   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Rel wrel 5694   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  cmpo 7433  1oc1o 8498  2oc2o 8499   Er wer 8741  [cec 8742  0cc0 11153  1c1 11154  cmin 11490  cn 12264  ...cfz 13544  ..^cfzo 13691  chash 14366  Word cword 14549   ++ cconcat 14605  ⟨“cs1 14630   splice csplice 14784  ⟨“cs2 14877   ~FG cefg 19739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-ec 8746  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-splice 14785  df-s2 14884  df-efg 19742
This theorem is referenced by:  efgrelex  19784
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