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Theorem efgrelexlemb 18361
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 18335 . 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 5399 . . . . . . 7 Rel 𝐿
87a1i 11 . . . . . 6 (⊤ → Rel 𝐿)
9 simpr 479 . . . . . . 7 ((⊤ ∧ 𝑓𝐿𝑔) → 𝑓𝐿𝑔)
10 eqcom 2765 . . . . . . . . . 10 ((𝑎‘0) = (𝑏‘0) ↔ (𝑏‘0) = (𝑎‘0))
11102rexbii 3178 . . . . . . . . 9 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0))
12 rexcom 3235 . . . . . . . . 9 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0) ↔ ∃𝑏 ∈ (𝑆 “ {𝑔})∃𝑎 ∈ (𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
1311, 12bitri 264 . . . . . . . 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 18360 . . . . . . . 8 (𝑓𝐿𝑔 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0))
171, 2, 3, 4, 14, 15, 6efgrelexlema 18360 . . . . . . . 8 (𝑔𝐿𝑓 ↔ ∃𝑏 ∈ (𝑆 “ {𝑔})∃𝑎 ∈ (𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
1813, 16, 173bitr4i 292 . . . . . . 7 (𝑓𝐿𝑔𝑔𝐿𝑓)
199, 18sylib 208 . . . . . 6 ((⊤ ∧ 𝑓𝐿𝑔) → 𝑔𝐿𝑓)
201, 2, 3, 4, 14, 15, 6efgrelexlema 18360 . . . . . . . . 9 (𝑔𝐿 ↔ ∃𝑟 ∈ (𝑆 “ {𝑔})∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0))
21 reeanv 3243 . . . . . . . . . 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 18350 . . . . . . . . . . . . . . . . . . . 20 𝑆:dom 𝑆onto𝑊
23 fofn 6276 . . . . . . . . . . . . . . . . . . . 20 (𝑆:dom 𝑆onto𝑊𝑆 Fn dom 𝑆)
2422, 23ax-mp 5 . . . . . . . . . . . . . . . . . . 19 𝑆 Fn dom 𝑆
25 fniniseg 6499 . . . . . . . . . . . . . . . . . . 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 6499 . . . . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . . . . . . 20 (((𝑆𝑟) = 𝑔 ∧ (𝑆𝑏) = 𝑔) → (𝑆𝑟) = (𝑆𝑏))
301, 2, 3, 4, 14, 15efgred 18359 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆 ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑟‘0) = (𝑏‘0))
3130eqcomd 2764 . . . . . . . . . . . . . . . . . . . . 21 ((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆 ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑏‘0) = (𝑟‘0))
32313expa 1112 . . . . . . . . . . . . . . . . . . . 20 (((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆) ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑏‘0) = (𝑟‘0))
3329, 32sylan2 492 . . . . . . . . . . . . . . . . . . 19 (((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆) ∧ ((𝑆𝑟) = 𝑔 ∧ (𝑆𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
3433an4s 904 . . . . . . . . . . . . . . . . . 18 (((𝑟 ∈ dom 𝑆 ∧ (𝑆𝑟) = 𝑔) ∧ (𝑏 ∈ dom 𝑆 ∧ (𝑆𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
3526, 28, 34syl2anb 497 . . . . . . . . . . . . . . . . 17 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → (𝑏‘0) = (𝑟‘0))
36 eqeq2 2769 . . . . . . . . . . . . . . . . 17 ((𝑟‘0) = (𝑠‘0) → ((𝑏‘0) = (𝑟‘0) ↔ (𝑏‘0) = (𝑠‘0)))
3735, 36syl5ibcom 235 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → ((𝑟‘0) = (𝑠‘0) → (𝑏‘0) = (𝑠‘0)))
3837reximdv 3152 . . . . . . . . . . . . . . 15 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑏‘0) = (𝑠‘0)))
39 eqeq1 2762 . . . . . . . . . . . . . . . . 17 ((𝑎‘0) = (𝑏‘0) → ((𝑎‘0) = (𝑠‘0) ↔ (𝑏‘0) = (𝑠‘0)))
4039rexbidv 3188 . . . . . . . . . . . . . . . 16 ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0) ↔ ∃𝑠 ∈ (𝑆 “ {})(𝑏‘0) = (𝑠‘0)))
4140imbi2d 329 . . . . . . . . . . . . . . 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 237 . . . . . . . . . . . . . 14 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))))
4342rexlimdva 3167 . . . . . . . . . . . . 13 (𝑟 ∈ (𝑆 “ {𝑔}) → (∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))))
4443impd 446 . . . . . . . . . . . 12 (𝑟 ∈ (𝑆 “ {𝑔}) → ((∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0)))
4544rexlimiv 3163 . . . . . . . . . . 11 (∃𝑟 ∈ (𝑆 “ {𝑔})(∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
4645reximi 3147 . . . . . . . . . 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 225 . . . . . . . . 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 497 . . . . . . . 8 ((𝑓𝐿𝑔𝑔𝐿) → ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
491, 2, 3, 4, 14, 15, 6efgrelexlema 18360 . . . . . . . 8 (𝑓𝐿 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
5048, 49sylibr 224 . . . . . . 7 ((𝑓𝐿𝑔𝑔𝐿) → 𝑓𝐿)
5150adantl 473 . . . . . 6 ((⊤ ∧ (𝑓𝐿𝑔𝑔𝐿)) → 𝑓𝐿)
52 eqid 2758 . . . . . . . . . . . 12 (𝑎‘0) = (𝑎‘0)
53 fveq1 6349 . . . . . . . . . . . . . 14 (𝑏 = 𝑎 → (𝑏‘0) = (𝑎‘0))
5453eqeq2d 2768 . . . . . . . . . . . . 13 (𝑏 = 𝑎 → ((𝑎‘0) = (𝑏‘0) ↔ (𝑎‘0) = (𝑎‘0)))
5554rspcev 3447 . . . . . . . . . . . 12 ((𝑎 ∈ (𝑆 “ {𝑓}) ∧ (𝑎‘0) = (𝑎‘0)) → ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
5652, 55mpan2 709 . . . . . . . . . . 11 (𝑎 ∈ (𝑆 “ {𝑓}) → ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
5756pm4.71i 667 . . . . . . . . . 10 (𝑎 ∈ (𝑆 “ {𝑓}) ↔ (𝑎 ∈ (𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)))
58 fniniseg 6499 . . . . . . . . . . 11 (𝑆 Fn dom 𝑆 → (𝑎 ∈ (𝑆 “ {𝑓}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆𝑎) = 𝑓)))
5924, 58ax-mp 5 . . . . . . . . . 10 (𝑎 ∈ (𝑆 “ {𝑓}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆𝑎) = 𝑓))
6057, 59bitr3i 266 . . . . . . . . 9 ((𝑎 ∈ (𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆𝑎) = 𝑓))
6160rexbii2 3175 . . . . . . . 8 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓)
621, 2, 3, 4, 14, 15, 6efgrelexlema 18360 . . . . . . . 8 (𝑓𝐿𝑓 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
63 forn 6277 . . . . . . . . . . 11 (𝑆:dom 𝑆onto𝑊 → ran 𝑆 = 𝑊)
6422, 63ax-mp 5 . . . . . . . . . 10 ran 𝑆 = 𝑊
6564eleq2i 2829 . . . . . . . . 9 (𝑓 ∈ ran 𝑆𝑓𝑊)
66 fvelrnb 6403 . . . . . . . . . 10 (𝑆 Fn dom 𝑆 → (𝑓 ∈ ran 𝑆 ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓))
6724, 66ax-mp 5 . . . . . . . . 9 (𝑓 ∈ ran 𝑆 ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓)
6865, 67bitr3i 266 . . . . . . . 8 (𝑓𝑊 ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓)
6961, 62, 683bitr4ri 293 . . . . . . 7 (𝑓𝑊𝑓𝐿𝑓)
7069a1i 11 . . . . . 6 (⊤ → (𝑓𝑊𝑓𝐿𝑓))
718, 19, 51, 70iserd 7935 . . . . 5 (⊤ → 𝐿 Er 𝑊)
7271trud 1640 . . . 4 𝐿 Er 𝑊
73 simpl 474 . . . . . . . . . . 11 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑎𝑊)
74 foelrn 6539 . . . . . . . . . . 11 ((𝑆:dom 𝑆onto𝑊𝑎𝑊) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆𝑟))
7522, 73, 74sylancr 698 . . . . . . . . . 10 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆𝑟))
76 simprl 811 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ dom 𝑆)
77 simprr 813 . . . . . . . . . . . . . . 15 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑎 = (𝑆𝑟))
7877eqcomd 2764 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑆𝑟) = 𝑎)
79 fniniseg 6499 . . . . . . . . . . . . . . 15 (𝑆 Fn dom 𝑆 → (𝑟 ∈ (𝑆 “ {𝑎}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆𝑟) = 𝑎)))
8024, 79ax-mp 5 . . . . . . . . . . . . . 14 (𝑟 ∈ (𝑆 “ {𝑎}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆𝑟) = 𝑎))
8176, 78, 80sylanbrc 701 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ (𝑆 “ {𝑎}))
82 simplr 809 . . . . . . . . . . . . . . . . 17 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏 ∈ ran (𝑇𝑎))
8377fveq2d 6354 . . . . . . . . . . . . . . . . . 18 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑇𝑎) = (𝑇‘(𝑆𝑟)))
8483rneqd 5506 . . . . . . . . . . . . . . . . 17 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ran (𝑇𝑎) = ran (𝑇‘(𝑆𝑟)))
8582, 84eleqtrd 2839 . . . . . . . . . . . . . . . 16 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏 ∈ ran (𝑇‘(𝑆𝑟)))
861, 2, 3, 4, 14, 15efgsp1 18348 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ dom 𝑆𝑏 ∈ ran (𝑇‘(𝑆𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
8776, 85, 86syl2anc 696 . . . . . . . . . . . . . . 15 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
881, 2, 3, 4, 14, 15efgsdm 18341 . . . . . . . . . . . . . . . . . . 19 (𝑟 ∈ dom 𝑆 ↔ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑟‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑟))(𝑟𝑖) ∈ ran (𝑇‘(𝑟‘(𝑖 − 1)))))
8988simp1bi 1140 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ dom 𝑆𝑟 ∈ (Word 𝑊 ∖ {∅}))
9089ad2antrl 766 . . . . . . . . . . . . . . . . 17 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
9190eldifad 3725 . . . . . . . . . . . . . . . 16 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ Word 𝑊)
921, 2, 3, 4efgtf 18333 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝑊 → ((𝑇𝑎) = (𝑓 ∈ (0...(♯‘𝑎)), 𝑔 ∈ (𝐼 × 2𝑜) ↦ (𝑎 splice ⟨𝑓, 𝑓, ⟨“𝑔(𝑀𝑔)”⟩⟩)) ∧ (𝑇𝑎):((0...(♯‘𝑎)) × (𝐼 × 2𝑜))⟶𝑊))
9392simprd 482 . . . . . . . . . . . . . . . . . . 19 (𝑎𝑊 → (𝑇𝑎):((0...(♯‘𝑎)) × (𝐼 × 2𝑜))⟶𝑊)
94 frn 6212 . . . . . . . . . . . . . . . . . . 19 ((𝑇𝑎):((0...(♯‘𝑎)) × (𝐼 × 2𝑜))⟶𝑊 → ran (𝑇𝑎) ⊆ 𝑊)
9593, 94syl 17 . . . . . . . . . . . . . . . . . 18 (𝑎𝑊 → ran (𝑇𝑎) ⊆ 𝑊)
9695sselda 3742 . . . . . . . . . . . . . . . . 17 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑏𝑊)
9796adantr 472 . . . . . . . . . . . . . . . 16 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏𝑊)
981, 2, 3, 4, 14, 15efgsval2 18344 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ Word 𝑊𝑏𝑊 ∧ (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
9991, 97, 87, 98syl3anc 1477 . . . . . . . . . . . . . . 15 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
100 fniniseg 6499 . . . . . . . . . . . . . . . 16 (𝑆 Fn dom 𝑆 → ((𝑟 ++ ⟨“𝑏”⟩) ∈ (𝑆 “ {𝑏}) ↔ ((𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆 ∧ (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)))
10124, 100ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑟 ++ ⟨“𝑏”⟩) ∈ (𝑆 “ {𝑏}) ↔ ((𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆 ∧ (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏))
10287, 99, 101sylanbrc 701 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ (𝑆 “ {𝑏}))
10397s1cld 13571 . . . . . . . . . . . . . . . 16 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ⟨“𝑏”⟩ ∈ Word 𝑊)
104 eldifsn 4460 . . . . . . . . . . . . . . . . . . 19 (𝑟 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝑟 ∈ Word 𝑊𝑟 ≠ ∅))
105 lennncl 13509 . . . . . . . . . . . . . . . . . . 19 ((𝑟 ∈ Word 𝑊𝑟 ≠ ∅) → (♯‘𝑟) ∈ ℕ)
106104, 105sylbi 207 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ (Word 𝑊 ∖ {∅}) → (♯‘𝑟) ∈ ℕ)
10790, 106syl 17 . . . . . . . . . . . . . . . . 17 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (♯‘𝑟) ∈ ℕ)
108 lbfzo0 12700 . . . . . . . . . . . . . . . . 17 (0 ∈ (0..^(♯‘𝑟)) ↔ (♯‘𝑟) ∈ ℕ)
109107, 108sylibr 224 . . . . . . . . . . . . . . . 16 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 0 ∈ (0..^(♯‘𝑟)))
110 ccatval1 13547 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ Word 𝑊 ∧ ⟨“𝑏”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(♯‘𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
11191, 103, 109, 110syl3anc 1477 . . . . . . . . . . . . . . 15 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
112111eqcomd 2764 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
113 fveq1 6349 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑟 ++ ⟨“𝑏”⟩) → (𝑠‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
114113eqeq2d 2768 . . . . . . . . . . . . . . 15 (𝑠 = (𝑟 ++ ⟨“𝑏”⟩) → ((𝑟‘0) = (𝑠‘0) ↔ (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0)))
115114rspcev 3447 . . . . . . . . . . . . . 14 (((𝑟 ++ ⟨“𝑏”⟩) ∈ (𝑆 “ {𝑏}) ∧ (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0)) → ∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
116102, 112, 115syl2anc 696 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
11781, 116jca 555 . . . . . . . . . . . 12 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟 ∈ (𝑆 “ {𝑎}) ∧ ∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0)))
118117ex 449 . . . . . . . . . . 11 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → ((𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟)) → (𝑟 ∈ (𝑆 “ {𝑎}) ∧ ∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))))
119118reximdv2 3150 . . . . . . . . . 10 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → (∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆𝑟) → ∃𝑟 ∈ (𝑆 “ {𝑎})∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0)))
12075, 119mpd 15 . . . . . . . . 9 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → ∃𝑟 ∈ (𝑆 “ {𝑎})∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
1211, 2, 3, 4, 14, 15, 6efgrelexlema 18360 . . . . . . . . 9 (𝑎𝐿𝑏 ↔ ∃𝑟 ∈ (𝑆 “ {𝑎})∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
122120, 121sylibr 224 . . . . . . . 8 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑎𝐿𝑏)
123 vex 3341 . . . . . . . . 9 𝑏 ∈ V
124 vex 3341 . . . . . . . . 9 𝑎 ∈ V
125123, 124elec 7951 . . . . . . . 8 (𝑏 ∈ [𝑎]𝐿𝑎𝐿𝑏)
126122, 125sylibr 224 . . . . . . 7 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑏 ∈ [𝑎]𝐿)
127126ex 449 . . . . . 6 (𝑎𝑊 → (𝑏 ∈ ran (𝑇𝑎) → 𝑏 ∈ [𝑎]𝐿))
128127ssrdv 3748 . . . . 5 (𝑎𝑊 → ran (𝑇𝑎) ⊆ [𝑎]𝐿)
129128rgen 3058 . . . 4 𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿
130 fvex 6360 . . . . . . 7 ( I ‘Word (𝐼 × 2𝑜)) ∈ V
1311, 130eqeltri 2833 . . . . . 6 𝑊 ∈ V
132 erex 7933 . . . . . 6 (𝐿 Er 𝑊 → (𝑊 ∈ V → 𝐿 ∈ V))
13372, 131, 132mp2 9 . . . . 5 𝐿 ∈ V
134 ereq1 7916 . . . . . 6 (𝑟 = 𝐿 → (𝑟 Er 𝑊𝐿 Er 𝑊))
135 eceq2 7949 . . . . . . . 8 (𝑟 = 𝐿 → [𝑎]𝑟 = [𝑎]𝐿)
136135sseq2d 3772 . . . . . . 7 (𝑟 = 𝐿 → (ran (𝑇𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇𝑎) ⊆ [𝑎]𝐿))
137136ralbidv 3122 . . . . . 6 (𝑟 = 𝐿 → (∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟 ↔ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿))
138134, 137anbi12d 749 . . . . 5 (𝑟 = 𝐿 → ((𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟) ↔ (𝐿 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿)))
139133, 138elab 3488 . . . 4 (𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} ↔ (𝐿 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿))
14072, 129, 139mpbir2an 993 . . 3 𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)}
141 intss1 4642 . . 3 (𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} → {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} ⊆ 𝐿)
142140, 141ax-mp 5 . 2 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} ⊆ 𝐿
1435, 142eqsstri 3774 1 𝐿
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1630  wtru 1631  wcel 2137  {cab 2744  wne 2930  wral 3048  wrex 3049  {crab 3052  Vcvv 3338  cdif 3710  wss 3713  c0 4056  {csn 4319  cop 4325  cotp 4327   cint 4625   ciun 4670   class class class wbr 4802  {copab 4862  cmpt 4879   I cid 5171   × cxp 5262  ccnv 5263  dom cdm 5264  ran crn 5265  cima 5267  Rel wrel 5269   Fn wfn 6042  wf 6043  ontowfo 6045  cfv 6047  (class class class)co 6811  cmpt2 6813  1𝑜c1o 7720  2𝑜c2o 7721   Er wer 7906  [cec 7907  0cc0 10126  1c1 10127  cmin 10456  cn 11210  ...cfz 12517  ..^cfzo 12657  chash 13309  Word cword 13475   ++ cconcat 13477  ⟨“cs1 13478   splice csplice 13480  ⟨“cs2 13784   ~FG cefg 18317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-ot 4328  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-er 7909  df-ec 7911  df-map 8023  df-pm 8024  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-card 8953  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-nn 11211  df-2 11269  df-n0 11483  df-z 11568  df-uz 11878  df-rp 12024  df-fz 12518  df-fzo 12658  df-hash 13310  df-word 13483  df-concat 13485  df-s1 13486  df-substr 13487  df-splice 13488  df-s2 13791  df-efg 18320
This theorem is referenced by:  efgrelex  18362
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