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Theorem wwlktovf1 14908
Description: Lemma 2 for wrd2f1tovbij 14911. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
Hypotheses
Ref Expression
wwlktovf1o.d 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
wwlktovf1o.r 𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}
wwlktovf1o.f 𝐹 = (𝑡𝐷 ↦ (𝑡‘1))
Assertion
Ref Expression
wwlktovf1 𝐹:𝐷1-1𝑅
Distinct variable groups:   𝑡,𝐷   𝑃,𝑛,𝑡,𝑤   𝑡,𝑅   𝑛,𝑉,𝑡,𝑤   𝑛,𝑋,𝑤
Allowed substitution hints:   𝐷(𝑤,𝑛)   𝑅(𝑤,𝑛)   𝐹(𝑤,𝑡,𝑛)   𝑋(𝑡)

Proof of Theorem wwlktovf1
Dummy variables 𝑥 𝑦 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlktovf1o.d . . 3 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
2 wwlktovf1o.r . . 3 𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}
3 wwlktovf1o.f . . 3 𝐹 = (𝑡𝐷 ↦ (𝑡‘1))
41, 2, 3wwlktovf 14907 . 2 𝐹:𝐷𝑅
5 fveq1 6891 . . . . . 6 (𝑡 = 𝑥 → (𝑡‘1) = (𝑥‘1))
6 fvex 6905 . . . . . 6 (𝑥‘1) ∈ V
75, 3, 6fvmpt 6999 . . . . 5 (𝑥𝐷 → (𝐹𝑥) = (𝑥‘1))
8 fveq1 6891 . . . . . 6 (𝑡 = 𝑦 → (𝑡‘1) = (𝑦‘1))
9 fvex 6905 . . . . . 6 (𝑦‘1) ∈ V
108, 3, 9fvmpt 6999 . . . . 5 (𝑦𝐷 → (𝐹𝑦) = (𝑦‘1))
117, 10eqeqan12d 2747 . . . 4 ((𝑥𝐷𝑦𝐷) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝑥‘1) = (𝑦‘1)))
12 fveqeq2 6901 . . . . . . 7 (𝑤 = 𝑥 → ((♯‘𝑤) = 2 ↔ (♯‘𝑥) = 2))
13 fveq1 6891 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
1413eqeq1d 2735 . . . . . . 7 (𝑤 = 𝑥 → ((𝑤‘0) = 𝑃 ↔ (𝑥‘0) = 𝑃))
15 fveq1 6891 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤‘1) = (𝑥‘1))
1613, 15preq12d 4746 . . . . . . . 8 (𝑤 = 𝑥 → {(𝑤‘0), (𝑤‘1)} = {(𝑥‘0), (𝑥‘1)})
1716eleq1d 2819 . . . . . . 7 (𝑤 = 𝑥 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋))
1812, 14, 173anbi123d 1437 . . . . . 6 (𝑤 = 𝑥 → (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
1918, 1elrab2 3687 . . . . 5 (𝑥𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
20 fveqeq2 6901 . . . . . . 7 (𝑤 = 𝑦 → ((♯‘𝑤) = 2 ↔ (♯‘𝑦) = 2))
21 fveq1 6891 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
2221eqeq1d 2735 . . . . . . 7 (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
23 fveq1 6891 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤‘1) = (𝑦‘1))
2421, 23preq12d 4746 . . . . . . . 8 (𝑤 = 𝑦 → {(𝑤‘0), (𝑤‘1)} = {(𝑦‘0), (𝑦‘1)})
2524eleq1d 2819 . . . . . . 7 (𝑤 = 𝑦 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))
2620, 22, 253anbi123d 1437 . . . . . 6 (𝑤 = 𝑦 → (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
2726, 1elrab2 3687 . . . . 5 (𝑦𝐷 ↔ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
28 simpr1 1195 . . . . . . . . 9 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (♯‘𝑥) = 2)
29 simpr1 1195 . . . . . . . . . 10 ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → (♯‘𝑦) = 2)
3029eqcomd 2739 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 2 = (♯‘𝑦))
3128, 30sylan9eq 2793 . . . . . . . 8 (((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → (♯‘𝑥) = (♯‘𝑦))
3231adantr 482 . . . . . . 7 ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (♯‘𝑥) = (♯‘𝑦))
33 simpr2 1196 . . . . . . . . . 10 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (𝑥‘0) = 𝑃)
34 simpr2 1196 . . . . . . . . . . 11 ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → (𝑦‘0) = 𝑃)
3534eqcomd 2739 . . . . . . . . . 10 ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 𝑃 = (𝑦‘0))
3633, 35sylan9eq 2793 . . . . . . . . 9 (((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → (𝑥‘0) = (𝑦‘0))
3736adantr 482 . . . . . . . 8 ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥‘0) = (𝑦‘0))
38 simpr 486 . . . . . . . 8 ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥‘1) = (𝑦‘1))
39 oveq2 7417 . . . . . . . . . . . . 13 ((♯‘𝑥) = 2 → (0..^(♯‘𝑥)) = (0..^2))
40 fzo0to2pr 13717 . . . . . . . . . . . . 13 (0..^2) = {0, 1}
4139, 40eqtrdi 2789 . . . . . . . . . . . 12 ((♯‘𝑥) = 2 → (0..^(♯‘𝑥)) = {0, 1})
4241raleqdv 3326 . . . . . . . . . . 11 ((♯‘𝑥) = 2 → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ∀𝑖 ∈ {0, 1} (𝑥𝑖) = (𝑦𝑖)))
43 c0ex 11208 . . . . . . . . . . . 12 0 ∈ V
44 1ex 11210 . . . . . . . . . . . 12 1 ∈ V
45 fveq2 6892 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑥𝑖) = (𝑥‘0))
46 fveq2 6892 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑦𝑖) = (𝑦‘0))
4745, 46eqeq12d 2749 . . . . . . . . . . . 12 (𝑖 = 0 → ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
48 fveq2 6892 . . . . . . . . . . . . 13 (𝑖 = 1 → (𝑥𝑖) = (𝑥‘1))
49 fveq2 6892 . . . . . . . . . . . . 13 (𝑖 = 1 → (𝑦𝑖) = (𝑦‘1))
5048, 49eqeq12d 2749 . . . . . . . . . . . 12 (𝑖 = 1 → ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑥‘1) = (𝑦‘1)))
5143, 44, 47, 50ralpr 4705 . . . . . . . . . . 11 (∀𝑖 ∈ {0, 1} (𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1)))
5242, 51bitrdi 287 . . . . . . . . . 10 ((♯‘𝑥) = 2 → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
53523ad2ant1 1134 . . . . . . . . 9 (((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋) → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
5453ad3antlr 730 . . . . . . . 8 ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
5537, 38, 54mpbir2and 712 . . . . . . 7 ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → ∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖))
56 eqwrd 14507 . . . . . . . . 9 ((𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖))))
5756ad2ant2r 746 . . . . . . . 8 (((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖))))
5857adantr 482 . . . . . . 7 ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖))))
5932, 55, 58mpbir2and 712 . . . . . 6 ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → 𝑥 = 𝑦)
6059ex 414 . . . . 5 (((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → ((𝑥‘1) = (𝑦‘1) → 𝑥 = 𝑦))
6119, 27, 60syl2anb 599 . . . 4 ((𝑥𝐷𝑦𝐷) → ((𝑥‘1) = (𝑦‘1) → 𝑥 = 𝑦))
6211, 61sylbid 239 . . 3 ((𝑥𝐷𝑦𝐷) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6362rgen2 3198 . 2 𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
64 dff13 7254 . 2 (𝐹:𝐷1-1𝑅 ↔ (𝐹:𝐷𝑅 ∧ ∀𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
654, 63, 64mpbir2an 710 1 𝐹:𝐷1-1𝑅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  {crab 3433  {cpr 4631  cmpt 5232  wf 6540  1-1wf1 6541  cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111  2c2 12267  ..^cfzo 13627  chash 14290  Word cword 14464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465
This theorem is referenced by:  wwlktovf1o  14910
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