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Theorem efgrelexlemb 17986
 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 17960 . 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 5167 . . . . . . 7 Rel 𝐿
87a1i 11 . . . . . 6 (⊤ → Rel 𝐿)
9 simpr 476 . . . . . . 7 ((⊤ ∧ 𝑓𝐿𝑔) → 𝑓𝐿𝑔)
10 eqcom 2617 . . . . . . . . . 10 ((𝑎‘0) = (𝑏‘0) ↔ (𝑏‘0) = (𝑎‘0))
11102rexbii 3024 . . . . . . . . 9 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0))
12 rexcom 3080 . . . . . . . . 9 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0) ↔ ∃𝑏 ∈ (𝑆 “ {𝑔})∃𝑎 ∈ (𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
1311, 12bitri 263 . . . . . . . 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 17985 . . . . . . . 8 (𝑓𝐿𝑔 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0))
171, 2, 3, 4, 14, 15, 6efgrelexlema 17985 . . . . . . . 8 (𝑔𝐿𝑓 ↔ ∃𝑏 ∈ (𝑆 “ {𝑔})∃𝑎 ∈ (𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
1813, 16, 173bitr4i 291 . . . . . . 7 (𝑓𝐿𝑔𝑔𝐿𝑓)
199, 18sylib 207 . . . . . 6 ((⊤ ∧ 𝑓𝐿𝑔) → 𝑔𝐿𝑓)
201, 2, 3, 4, 14, 15, 6efgrelexlema 17985 . . . . . . . . 9 (𝑔𝐿 ↔ ∃𝑟 ∈ (𝑆 “ {𝑔})∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0))
21 reeanv 3086 . . . . . . . . . 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 17975 . . . . . . . . . . . . . . . . . . . 20 𝑆:dom 𝑆onto𝑊
23 fofn 6030 . . . . . . . . . . . . . . . . . . . 20 (𝑆:dom 𝑆onto𝑊𝑆 Fn dom 𝑆)
2422, 23ax-mp 5 . . . . . . . . . . . . . . . . . . 19 𝑆 Fn dom 𝑆
25 fniniseg 6246 . . . . . . . . . . . . . . . . . . 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 6246 . . . . . . . . . . . . . . . . . . 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 2631 . . . . . . . . . . . . . . . . . . . 20 (((𝑆𝑟) = 𝑔 ∧ (𝑆𝑏) = 𝑔) → (𝑆𝑟) = (𝑆𝑏))
301, 2, 3, 4, 14, 15efgred 17984 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆 ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑟‘0) = (𝑏‘0))
3130eqcomd 2616 . . . . . . . . . . . . . . . . . . . . 21 ((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆 ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑏‘0) = (𝑟‘0))
32313expa 1257 . . . . . . . . . . . . . . . . . . . 20 (((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆) ∧ (𝑆𝑟) = (𝑆𝑏)) → (𝑏‘0) = (𝑟‘0))
3329, 32sylan2 490 . . . . . . . . . . . . . . . . . . 19 (((𝑟 ∈ dom 𝑆𝑏 ∈ dom 𝑆) ∧ ((𝑆𝑟) = 𝑔 ∧ (𝑆𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
3433an4s 865 . . . . . . . . . . . . . . . . . 18 (((𝑟 ∈ dom 𝑆 ∧ (𝑆𝑟) = 𝑔) ∧ (𝑏 ∈ dom 𝑆 ∧ (𝑆𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
3526, 28, 34syl2anb 495 . . . . . . . . . . . . . . . . 17 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → (𝑏‘0) = (𝑟‘0))
36 eqeq2 2621 . . . . . . . . . . . . . . . . 17 ((𝑟‘0) = (𝑠‘0) → ((𝑏‘0) = (𝑟‘0) ↔ (𝑏‘0) = (𝑠‘0)))
3735, 36syl5ibcom 234 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → ((𝑟‘0) = (𝑠‘0) → (𝑏‘0) = (𝑠‘0)))
3837reximdv 2999 . . . . . . . . . . . . . . 15 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑏‘0) = (𝑠‘0)))
39 eqeq1 2614 . . . . . . . . . . . . . . . . 17 ((𝑎‘0) = (𝑏‘0) → ((𝑎‘0) = (𝑠‘0) ↔ (𝑏‘0) = (𝑠‘0)))
4039rexbidv 3034 . . . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . . 14 ((𝑟 ∈ (𝑆 “ {𝑔}) ∧ 𝑏 ∈ (𝑆 “ {𝑔})) → ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))))
4342rexlimdva 3013 . . . . . . . . . . . . 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 3009 . . . . . . . . . . 11 (∃𝑟 ∈ (𝑆 “ {𝑔})(∃𝑏 ∈ (𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (𝑆 “ {})(𝑟‘0) = (𝑠‘0)) → ∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
4645reximi 2994 . . . . . . . . . 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 224 . . . . . . . . 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 495 . . . . . . . 8 ((𝑓𝐿𝑔𝑔𝐿) → ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
491, 2, 3, 4, 14, 15, 6efgrelexlema 17985 . . . . . . . 8 (𝑓𝐿 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑠 ∈ (𝑆 “ {})(𝑎‘0) = (𝑠‘0))
5048, 49sylibr 223 . . . . . . 7 ((𝑓𝐿𝑔𝑔𝐿) → 𝑓𝐿)
5150adantl 481 . . . . . 6 ((⊤ ∧ (𝑓𝐿𝑔𝑔𝐿)) → 𝑓𝐿)
52 eqid 2610 . . . . . . . . . . . 12 (𝑎‘0) = (𝑎‘0)
53 fveq1 6102 . . . . . . . . . . . . . 14 (𝑏 = 𝑎 → (𝑏‘0) = (𝑎‘0))
5453eqeq2d 2620 . . . . . . . . . . . . 13 (𝑏 = 𝑎 → ((𝑎‘0) = (𝑏‘0) ↔ (𝑎‘0) = (𝑎‘0)))
5554rspcev 3282 . . . . . . . . . . . 12 ((𝑎 ∈ (𝑆 “ {𝑓}) ∧ (𝑎‘0) = (𝑎‘0)) → ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
5652, 55mpan2 703 . . . . . . . . . . 11 (𝑎 ∈ (𝑆 “ {𝑓}) → ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
5756pm4.71i 662 . . . . . . . . . 10 (𝑎 ∈ (𝑆 “ {𝑓}) ↔ (𝑎 ∈ (𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)))
58 fniniseg 6246 . . . . . . . . . . 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 265 . . . . . . . . 9 ((𝑎 ∈ (𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆𝑎) = 𝑓))
6160rexbii2 3021 . . . . . . . 8 (∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓)
621, 2, 3, 4, 14, 15, 6efgrelexlema 17985 . . . . . . . 8 (𝑓𝐿𝑓 ↔ ∃𝑎 ∈ (𝑆 “ {𝑓})∃𝑏 ∈ (𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
63 forn 6031 . . . . . . . . . . 11 (𝑆:dom 𝑆onto𝑊 → ran 𝑆 = 𝑊)
6422, 63ax-mp 5 . . . . . . . . . 10 ran 𝑆 = 𝑊
6564eleq2i 2680 . . . . . . . . 9 (𝑓 ∈ ran 𝑆𝑓𝑊)
66 fvelrnb 6153 . . . . . . . . . 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 265 . . . . . . . 8 (𝑓𝑊 ↔ ∃𝑎 ∈ dom 𝑆(𝑆𝑎) = 𝑓)
6961, 62, 683bitr4ri 292 . . . . . . 7 (𝑓𝑊𝑓𝐿𝑓)
7069a1i 11 . . . . . 6 (⊤ → (𝑓𝑊𝑓𝐿𝑓))
718, 19, 51, 70iserd 7655 . . . . 5 (⊤ → 𝐿 Er 𝑊)
7271trud 1484 . . . 4 𝐿 Er 𝑊
73 simpl 472 . . . . . . . . . . 11 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑎𝑊)
74 foelrn 6286 . . . . . . . . . . 11 ((𝑆:dom 𝑆onto𝑊𝑎𝑊) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆𝑟))
7522, 73, 74sylancr 694 . . . . . . . . . 10 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆𝑟))
76 simprl 790 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ dom 𝑆)
77 simprr 792 . . . . . . . . . . . . . . 15 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑎 = (𝑆𝑟))
7877eqcomd 2616 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑆𝑟) = 𝑎)
79 fniniseg 6246 . . . . . . . . . . . . . . 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 695 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ (𝑆 “ {𝑎}))
82 simplr 788 . . . . . . . . . . . . . . . . 17 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏 ∈ ran (𝑇𝑎))
8377fveq2d 6107 . . . . . . . . . . . . . . . . . 18 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑇𝑎) = (𝑇‘(𝑆𝑟)))
8483rneqd 5274 . . . . . . . . . . . . . . . . 17 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ran (𝑇𝑎) = ran (𝑇‘(𝑆𝑟)))
8582, 84eleqtrd 2690 . . . . . . . . . . . . . . . 16 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏 ∈ ran (𝑇‘(𝑆𝑟)))
861, 2, 3, 4, 14, 15efgsp1 17973 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ dom 𝑆𝑏 ∈ ran (𝑇‘(𝑆𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
8776, 85, 86syl2anc 691 . . . . . . . . . . . . . . 15 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
881, 2, 3, 4, 14, 15efgsdm 17966 . . . . . . . . . . . . . . . . . . 19 (𝑟 ∈ dom 𝑆 ↔ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑟‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝑟))(𝑟𝑖) ∈ ran (𝑇‘(𝑟‘(𝑖 − 1)))))
8988simp1bi 1069 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ dom 𝑆𝑟 ∈ (Word 𝑊 ∖ {∅}))
9089ad2antrl 760 . . . . . . . . . . . . . . . . 17 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
9190eldifad 3552 . . . . . . . . . . . . . . . 16 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑟 ∈ Word 𝑊)
921, 2, 3, 4efgtf 17958 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝑊 → ((𝑇𝑎) = (𝑓 ∈ (0...(#‘𝑎)), 𝑔 ∈ (𝐼 × 2𝑜) ↦ (𝑎 splice ⟨𝑓, 𝑓, ⟨“𝑔(𝑀𝑔)”⟩⟩)) ∧ (𝑇𝑎):((0...(#‘𝑎)) × (𝐼 × 2𝑜))⟶𝑊))
9392simprd 478 . . . . . . . . . . . . . . . . . . 19 (𝑎𝑊 → (𝑇𝑎):((0...(#‘𝑎)) × (𝐼 × 2𝑜))⟶𝑊)
94 frn 5966 . . . . . . . . . . . . . . . . . . 19 ((𝑇𝑎):((0...(#‘𝑎)) × (𝐼 × 2𝑜))⟶𝑊 → ran (𝑇𝑎) ⊆ 𝑊)
9593, 94syl 17 . . . . . . . . . . . . . . . . . 18 (𝑎𝑊 → ran (𝑇𝑎) ⊆ 𝑊)
9695sselda 3568 . . . . . . . . . . . . . . . . 17 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑏𝑊)
9796adantr 480 . . . . . . . . . . . . . . . 16 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 𝑏𝑊)
981, 2, 3, 4, 14, 15efgsval2 17969 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ Word 𝑊𝑏𝑊 ∧ (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
9991, 97, 87, 98syl3anc 1318 . . . . . . . . . . . . . . 15 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
100 fniniseg 6246 . . . . . . . . . . . . . . . 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 695 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ (𝑆 “ {𝑏}))
10397s1cld 13236 . . . . . . . . . . . . . . . 16 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ⟨“𝑏”⟩ ∈ Word 𝑊)
104 eldifsn 4260 . . . . . . . . . . . . . . . . . . 19 (𝑟 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝑟 ∈ Word 𝑊𝑟 ≠ ∅))
105 lennncl 13180 . . . . . . . . . . . . . . . . . . 19 ((𝑟 ∈ Word 𝑊𝑟 ≠ ∅) → (#‘𝑟) ∈ ℕ)
106104, 105sylbi 206 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ (Word 𝑊 ∖ {∅}) → (#‘𝑟) ∈ ℕ)
10790, 106syl 17 . . . . . . . . . . . . . . . . 17 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (#‘𝑟) ∈ ℕ)
108 lbfzo0 12375 . . . . . . . . . . . . . . . . 17 (0 ∈ (0..^(#‘𝑟)) ↔ (#‘𝑟) ∈ ℕ)
109107, 108sylibr 223 . . . . . . . . . . . . . . . 16 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → 0 ∈ (0..^(#‘𝑟)))
110 ccatval1 13214 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ Word 𝑊 ∧ ⟨“𝑏”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(#‘𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
11191, 103, 109, 110syl3anc 1318 . . . . . . . . . . . . . . 15 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
112111eqcomd 2616 . . . . . . . . . . . . . 14 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
113 fveq1 6102 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑟 ++ ⟨“𝑏”⟩) → (𝑠‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
114113eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑠 = (𝑟 ++ ⟨“𝑏”⟩) → ((𝑟‘0) = (𝑠‘0) ↔ (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0)))
115114rspcev 3282 . . . . . . . . . . . . . 14 (((𝑟 ++ ⟨“𝑏”⟩) ∈ (𝑆 “ {𝑏}) ∧ (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0)) → ∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
116102, 112, 115syl2anc 691 . . . . . . . . . . . . 13 (((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) ∧ (𝑟 ∈ dom 𝑆𝑎 = (𝑆𝑟))) → ∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
11781, 116jca 553 . . . . . . . . . . . 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 2997 . . . . . . . . . 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 17985 . . . . . . . . 9 (𝑎𝐿𝑏 ↔ ∃𝑟 ∈ (𝑆 “ {𝑎})∃𝑠 ∈ (𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
122120, 121sylibr 223 . . . . . . . 8 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑎𝐿𝑏)
123 vex 3176 . . . . . . . . 9 𝑏 ∈ V
124 vex 3176 . . . . . . . . 9 𝑎 ∈ V
125123, 124elec 7673 . . . . . . . 8 (𝑏 ∈ [𝑎]𝐿𝑎𝐿𝑏)
126122, 125sylibr 223 . . . . . . 7 ((𝑎𝑊𝑏 ∈ ran (𝑇𝑎)) → 𝑏 ∈ [𝑎]𝐿)
127126ex 449 . . . . . 6 (𝑎𝑊 → (𝑏 ∈ ran (𝑇𝑎) → 𝑏 ∈ [𝑎]𝐿))
128127ssrdv 3574 . . . . 5 (𝑎𝑊 → ran (𝑇𝑎) ⊆ [𝑎]𝐿)
129128rgen 2906 . . . 4 𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿
130 fvex 6113 . . . . . . 7 ( I ‘Word (𝐼 × 2𝑜)) ∈ V
1311, 130eqeltri 2684 . . . . . 6 𝑊 ∈ V
132 erex 7653 . . . . . 6 (𝐿 Er 𝑊 → (𝑊 ∈ V → 𝐿 ∈ V))
13372, 131, 132mp2 9 . . . . 5 𝐿 ∈ V
134 ereq1 7636 . . . . . 6 (𝑟 = 𝐿 → (𝑟 Er 𝑊𝐿 Er 𝑊))
135 eceq2 7671 . . . . . . . 8 (𝑟 = 𝐿 → [𝑎]𝑟 = [𝑎]𝐿)
136135sseq2d 3596 . . . . . . 7 (𝑟 = 𝐿 → (ran (𝑇𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇𝑎) ⊆ [𝑎]𝐿))
137136ralbidv 2969 . . . . . 6 (𝑟 = 𝐿 → (∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟 ↔ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿))
138134, 137anbi12d 743 . . . . 5 (𝑟 = 𝐿 → ((𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟) ↔ (𝐿 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿)))
139133, 138elab 3319 . . . 4 (𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} ↔ (𝐿 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝐿))
14072, 129, 139mpbir2an 957 . . 3 𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)}
141 intss1 4427 . . 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 3598 1 𝐿
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ⟨cotp 4133  ∩ cint 4410  ∪ ciun 4455   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   I cid 4948   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  Rel wrel 5043   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1𝑜c1o 7440  2𝑜c2o 7441   Er wer 7626  [cec 7627  0cc0 9815  1c1 9816   − cmin 10145  ℕcn 10897  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149   splice csplice 13151  ⟨“cs2 13437   ~FG cefg 17942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-s2 13444  df-efg 17945 This theorem is referenced by:  efgrelex  17987
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