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Theorem wwlktovf1 14904
Description: Lemma 2 for wrd2f1tovbij 14907. (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 14903 . 2 𝐹:𝐷𝑅
5 fveq1 6887 . . . . . 6 (𝑡 = 𝑥 → (𝑡‘1) = (𝑥‘1))
6 fvex 6901 . . . . . 6 (𝑥‘1) ∈ V
75, 3, 6fvmpt 6995 . . . . 5 (𝑥𝐷 → (𝐹𝑥) = (𝑥‘1))
8 fveq1 6887 . . . . . 6 (𝑡 = 𝑦 → (𝑡‘1) = (𝑦‘1))
9 fvex 6901 . . . . . 6 (𝑦‘1) ∈ V
108, 3, 9fvmpt 6995 . . . . 5 (𝑦𝐷 → (𝐹𝑦) = (𝑦‘1))
117, 10eqeqan12d 2746 . . . 4 ((𝑥𝐷𝑦𝐷) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝑥‘1) = (𝑦‘1)))
12 fveqeq2 6897 . . . . . . 7 (𝑤 = 𝑥 → ((♯‘𝑤) = 2 ↔ (♯‘𝑥) = 2))
13 fveq1 6887 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
1413eqeq1d 2734 . . . . . . 7 (𝑤 = 𝑥 → ((𝑤‘0) = 𝑃 ↔ (𝑥‘0) = 𝑃))
15 fveq1 6887 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤‘1) = (𝑥‘1))
1613, 15preq12d 4744 . . . . . . . 8 (𝑤 = 𝑥 → {(𝑤‘0), (𝑤‘1)} = {(𝑥‘0), (𝑥‘1)})
1716eleq1d 2818 . . . . . . 7 (𝑤 = 𝑥 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋))
1812, 14, 173anbi123d 1436 . . . . . 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 3685 . . . . 5 (𝑥𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
20 fveqeq2 6897 . . . . . . 7 (𝑤 = 𝑦 → ((♯‘𝑤) = 2 ↔ (♯‘𝑦) = 2))
21 fveq1 6887 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
2221eqeq1d 2734 . . . . . . 7 (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
23 fveq1 6887 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤‘1) = (𝑦‘1))
2421, 23preq12d 4744 . . . . . . . 8 (𝑤 = 𝑦 → {(𝑤‘0), (𝑤‘1)} = {(𝑦‘0), (𝑦‘1)})
2524eleq1d 2818 . . . . . . 7 (𝑤 = 𝑦 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))
2620, 22, 253anbi123d 1436 . . . . . 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 3685 . . . . 5 (𝑦𝐷 ↔ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
28 simpr1 1194 . . . . . . . . 9 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (♯‘𝑥) = 2)
29 simpr1 1194 . . . . . . . . . 10 ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → (♯‘𝑦) = 2)
3029eqcomd 2738 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 2 = (♯‘𝑦))
3128, 30sylan9eq 2792 . . . . . . . 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 481 . . . . . . 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 1195 . . . . . . . . . 10 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (𝑥‘0) = 𝑃)
34 simpr2 1195 . . . . . . . . . . 11 ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → (𝑦‘0) = 𝑃)
3534eqcomd 2738 . . . . . . . . . 10 ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 𝑃 = (𝑦‘0))
3633, 35sylan9eq 2792 . . . . . . . . 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 481 . . . . . . . 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 485 . . . . . . . 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 7413 . . . . . . . . . . . . 13 ((♯‘𝑥) = 2 → (0..^(♯‘𝑥)) = (0..^2))
40 fzo0to2pr 13713 . . . . . . . . . . . . 13 (0..^2) = {0, 1}
4139, 40eqtrdi 2788 . . . . . . . . . . . 12 ((♯‘𝑥) = 2 → (0..^(♯‘𝑥)) = {0, 1})
4241raleqdv 3325 . . . . . . . . . . 11 ((♯‘𝑥) = 2 → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ∀𝑖 ∈ {0, 1} (𝑥𝑖) = (𝑦𝑖)))
43 c0ex 11204 . . . . . . . . . . . 12 0 ∈ V
44 1ex 11206 . . . . . . . . . . . 12 1 ∈ V
45 fveq2 6888 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑥𝑖) = (𝑥‘0))
46 fveq2 6888 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑦𝑖) = (𝑦‘0))
4745, 46eqeq12d 2748 . . . . . . . . . . . 12 (𝑖 = 0 → ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
48 fveq2 6888 . . . . . . . . . . . . 13 (𝑖 = 1 → (𝑥𝑖) = (𝑥‘1))
49 fveq2 6888 . . . . . . . . . . . . 13 (𝑖 = 1 → (𝑦𝑖) = (𝑦‘1))
5048, 49eqeq12d 2748 . . . . . . . . . . . 12 (𝑖 = 1 → ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑥‘1) = (𝑦‘1)))
5143, 44, 47, 50ralpr 4703 . . . . . . . . . . 11 (∀𝑖 ∈ {0, 1} (𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1)))
5242, 51bitrdi 286 . . . . . . . . . 10 ((♯‘𝑥) = 2 → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
53523ad2ant1 1133 . . . . . . . . 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 729 . . . . . . . 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 711 . . . . . . 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 14503 . . . . . . . . 9 ((𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖))))
5756ad2ant2r 745 . . . . . . . 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 481 . . . . . . 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 711 . . . . . 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 413 . . . . 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 598 . . . 4 ((𝑥𝐷𝑦𝐷) → ((𝑥‘1) = (𝑦‘1) → 𝑥 = 𝑦))
6211, 61sylbid 239 . . 3 ((𝑥𝐷𝑦𝐷) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6362rgen2 3197 . 2 𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
64 dff13 7250 . 2 (𝐹:𝐷1-1𝑅 ↔ (𝐹:𝐷𝑅 ∧ ∀𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
654, 63, 64mpbir2an 709 1 𝐹:𝐷1-1𝑅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  {crab 3432  {cpr 4629  cmpt 5230  wf 6536  1-1wf1 6537  cfv 6540  (class class class)co 7405  0cc0 11106  1c1 11107  2c2 12263  ..^cfzo 13623  chash 14286  Word cword 14460
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461
This theorem is referenced by:  wwlktovf1o  14906
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