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Theorem wwlktovf1 13921
Description: Lemma 2 for wrd2f1tovbij 13924. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
Hypotheses
Ref Expression
wrd2f1tovbij.d 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
wrd2f1tovbij.r 𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}
wrd2f1tovbij.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 wrd2f1tovbij.d . . 3 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
2 wrd2f1tovbij.r . . 3 𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}
3 wrd2f1tovbij.f . . 3 𝐹 = (𝑡𝐷 ↦ (𝑡‘1))
41, 2, 3wwlktovf 13920 . 2 𝐹:𝐷𝑅
5 fveq1 6352 . . . . . 6 (𝑡 = 𝑥 → (𝑡‘1) = (𝑥‘1))
6 fvex 6363 . . . . . 6 (𝑥‘1) ∈ V
75, 3, 6fvmpt 6445 . . . . 5 (𝑥𝐷 → (𝐹𝑥) = (𝑥‘1))
8 fveq1 6352 . . . . . 6 (𝑡 = 𝑦 → (𝑡‘1) = (𝑦‘1))
9 fvex 6363 . . . . . 6 (𝑦‘1) ∈ V
108, 3, 9fvmpt 6445 . . . . 5 (𝑦𝐷 → (𝐹𝑦) = (𝑦‘1))
117, 10eqeqan12d 2776 . . . 4 ((𝑥𝐷𝑦𝐷) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝑥‘1) = (𝑦‘1)))
12 fveq2 6353 . . . . . . . 8 (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
1312eqeq1d 2762 . . . . . . 7 (𝑤 = 𝑥 → ((♯‘𝑤) = 2 ↔ (♯‘𝑥) = 2))
14 fveq1 6352 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
1514eqeq1d 2762 . . . . . . 7 (𝑤 = 𝑥 → ((𝑤‘0) = 𝑃 ↔ (𝑥‘0) = 𝑃))
16 fveq1 6352 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤‘1) = (𝑥‘1))
1714, 16preq12d 4420 . . . . . . . 8 (𝑤 = 𝑥 → {(𝑤‘0), (𝑤‘1)} = {(𝑥‘0), (𝑥‘1)})
1817eleq1d 2824 . . . . . . 7 (𝑤 = 𝑥 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋))
1913, 15, 183anbi123d 1548 . . . . . 6 (𝑤 = 𝑥 → (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
2019, 1elrab2 3507 . . . . 5 (𝑥𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
21 fveq2 6353 . . . . . . . 8 (𝑤 = 𝑦 → (♯‘𝑤) = (♯‘𝑦))
2221eqeq1d 2762 . . . . . . 7 (𝑤 = 𝑦 → ((♯‘𝑤) = 2 ↔ (♯‘𝑦) = 2))
23 fveq1 6352 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
2423eqeq1d 2762 . . . . . . 7 (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
25 fveq1 6352 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤‘1) = (𝑦‘1))
2623, 25preq12d 4420 . . . . . . . 8 (𝑤 = 𝑦 → {(𝑤‘0), (𝑤‘1)} = {(𝑦‘0), (𝑦‘1)})
2726eleq1d 2824 . . . . . . 7 (𝑤 = 𝑦 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))
2822, 24, 273anbi123d 1548 . . . . . 6 (𝑤 = 𝑦 → (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
2928, 1elrab2 3507 . . . . 5 (𝑦𝐷 ↔ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
30 simp1 1131 . . . . . . . . . 10 (((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋) → (♯‘𝑥) = 2)
3130adantl 473 . . . . . . . . 9 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (♯‘𝑥) = 2)
32 simp1 1131 . . . . . . . . . . 11 (((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋) → (♯‘𝑦) = 2)
3332eqcomd 2766 . . . . . . . . . 10 (((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋) → 2 = (♯‘𝑦))
3433adantl 473 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 2 = (♯‘𝑦))
3531, 34sylan9eq 2814 . . . . . . . 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))
3635adantr 472 . . . . . . 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))
37 simp2 1132 . . . . . . . . . . 11 (((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋) → (𝑥‘0) = 𝑃)
3837adantl 473 . . . . . . . . . 10 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (𝑥‘0) = 𝑃)
39 simp2 1132 . . . . . . . . . . . 12 (((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋) → (𝑦‘0) = 𝑃)
4039eqcomd 2766 . . . . . . . . . . 11 (((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋) → 𝑃 = (𝑦‘0))
4140adantl 473 . . . . . . . . . 10 ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 𝑃 = (𝑦‘0))
4238, 41sylan9eq 2814 . . . . . . . . 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))
4342adantr 472 . . . . . . . 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))
44 simpr 479 . . . . . . . 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))
45 oveq2 6822 . . . . . . . . . . . . 13 ((♯‘𝑥) = 2 → (0..^(♯‘𝑥)) = (0..^2))
46 fzo0to2pr 12767 . . . . . . . . . . . . 13 (0..^2) = {0, 1}
4745, 46syl6eq 2810 . . . . . . . . . . . 12 ((♯‘𝑥) = 2 → (0..^(♯‘𝑥)) = {0, 1})
4847raleqdv 3283 . . . . . . . . . . 11 ((♯‘𝑥) = 2 → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ∀𝑖 ∈ {0, 1} (𝑥𝑖) = (𝑦𝑖)))
49 c0ex 10246 . . . . . . . . . . . 12 0 ∈ V
50 1ex 10247 . . . . . . . . . . . 12 1 ∈ V
51 fveq2 6353 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑥𝑖) = (𝑥‘0))
52 fveq2 6353 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑦𝑖) = (𝑦‘0))
5351, 52eqeq12d 2775 . . . . . . . . . . . 12 (𝑖 = 0 → ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
54 fveq2 6353 . . . . . . . . . . . . 13 (𝑖 = 1 → (𝑥𝑖) = (𝑥‘1))
55 fveq2 6353 . . . . . . . . . . . . 13 (𝑖 = 1 → (𝑦𝑖) = (𝑦‘1))
5654, 55eqeq12d 2775 . . . . . . . . . . . 12 (𝑖 = 1 → ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑥‘1) = (𝑦‘1)))
5749, 50, 53, 56ralpr 4382 . . . . . . . . . . 11 (∀𝑖 ∈ {0, 1} (𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1)))
5848, 57syl6bb 276 . . . . . . . . . 10 ((♯‘𝑥) = 2 → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
59583ad2ant1 1128 . . . . . . . . 9 (((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋) → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
6059ad3antlr 769 . . . . . . . 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))))
6143, 44, 60mpbir2and 995 . . . . . . 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))
62 simpl 474 . . . . . . . . . 10 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → 𝑥 ∈ Word 𝑉)
63 simpl 474 . . . . . . . . . 10 ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 𝑦 ∈ Word 𝑉)
6462, 63anim12i 591 . . . . . . . . 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 ∈ Word 𝑉𝑦 ∈ Word 𝑉))
6564adantr 472 . . . . . . . 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 ∈ Word 𝑉𝑦 ∈ Word 𝑉))
66 eqwrd 13553 . . . . . . . 8 ((𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) = (𝑦𝑖))))
6765, 66syl 17 . . . . . . 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))))
6836, 61, 67mpbir2and 995 . . . . . 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)
6968ex 449 . . . . 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))
7020, 29, 69syl2anb 497 . . . 4 ((𝑥𝐷𝑦𝐷) → ((𝑥‘1) = (𝑦‘1) → 𝑥 = 𝑦))
7111, 70sylbid 230 . . 3 ((𝑥𝐷𝑦𝐷) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
7271rgen2a 3115 . 2 𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
73 dff13 6676 . 2 (𝐹:𝐷1-1𝑅 ↔ (𝐹:𝐷𝑅 ∧ ∀𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
744, 72, 73mpbir2an 993 1 𝐹:𝐷1-1𝑅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  {crab 3054  {cpr 4323  cmpt 4881  wf 6045  1-1wf1 6046  cfv 6049  (class class class)co 6814  0cc0 10148  1c1 10149  2c2 11282  ..^cfzo 12679  chash 13331  Word cword 13497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-fzo 12680  df-hash 13332  df-word 13505
This theorem is referenced by:  wwlktovf1o  13923
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