Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wwlks2onv Structured version   Visualization version   GIF version

Theorem wwlks2onv 41158
 Description: If a length 3 string represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
Hypothesis
Ref Expression
wwlks2onv.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
wwlks2onv ((𝐵𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴𝑉𝐵𝑉𝐶𝑉))

Proof of Theorem wwlks2onv
Dummy variables 𝑎 𝑐 𝑤 𝑏 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2nn0 11186 . . . . . . . 8 2 ∈ ℕ0
2 wwlks2onv.v . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
32wwlksnon 41049 . . . . . . . 8 ((2 ∈ ℕ0𝐺 ∈ V) → (2 WWalksNOn 𝐺) = (𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)}))
41, 3mpan 702 . . . . . . 7 (𝐺 ∈ V → (2 WWalksNOn 𝐺) = (𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)}))
54oveqd 6566 . . . . . 6 (𝐺 ∈ V → (𝐴(2 WWalksNOn 𝐺)𝐶) = (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶))
65eleq2d 2673 . . . . 5 (𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶)))
7 eqid 2610 . . . . . . 7 (𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)}) = (𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})
87elmpt2cl 6774 . . . . . 6 (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐴𝑉𝐶𝑉))
9 simprl 790 . . . . . . . 8 (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵𝑈) ∧ (𝐴𝑉𝐶𝑉)) → 𝐴𝑉)
10 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴))
1110anbi1d 737 . . . . . . . . . . . . . 14 (𝑎 = 𝐴 → (((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝑐)))
1211rabbidv 3164 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)} = {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝑐)})
13 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑐 = 𝐶 → ((𝑤‘2) = 𝑐 ↔ (𝑤‘2) = 𝐶))
1413anbi2d 736 . . . . . . . . . . . . . 14 (𝑐 = 𝐶 → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝑐) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)))
1514rabbidv 3164 . . . . . . . . . . . . 13 (𝑐 = 𝐶 → {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝑐)} = {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)})
16 ovex 6577 . . . . . . . . . . . . . 14 (2 WWalkSN 𝐺) ∈ V
1716rabex 4740 . . . . . . . . . . . . 13 {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)} ∈ V
1812, 15, 7, 17ovmpt2 6694 . . . . . . . . . . . 12 ((𝐴𝑉𝐶𝑉) → (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) = {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)})
1918eleq2d 2673 . . . . . . . . . . 11 ((𝐴𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)}))
20 fveq1 6102 . . . . . . . . . . . . . . 15 (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → (𝑤‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
2120eqeq1d 2612 . . . . . . . . . . . . . 14 (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → ((𝑤‘0) = 𝐴 ↔ (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴))
22 fveq1 6102 . . . . . . . . . . . . . . 15 (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → (𝑤‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
2322eqeq1d 2612 . . . . . . . . . . . . . 14 (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → ((𝑤‘2) = 𝐶 ↔ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶))
2421, 23anbi12d 743 . . . . . . . . . . . . 13 (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶) ↔ ((⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)))
2524elrab 3331 . . . . . . . . . . . 12 (⟨“𝐴𝐵𝐶”⟩ ∈ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)} ↔ (⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalkSN 𝐺) ∧ ((⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)))
26 wwlknbp2 41063 . . . . . . . . . . . . . . 15 (⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalkSN 𝐺) → (⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)))
27 s3fv1 13487 . . . . . . . . . . . . . . . . . . 19 (𝐵𝑈 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
2827eqcomd 2616 . . . . . . . . . . . . . . . . . 18 (𝐵𝑈𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1))
2928adantl 481 . . . . . . . . . . . . . . . . 17 (((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) ∧ 𝐵𝑈) → 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1))
30 1ex 9914 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ V
3130tpid2 4247 . . . . . . . . . . . . . . . . . . . 20 1 ∈ {0, 1, 2}
32 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1))
33 2p1e3 11028 . . . . . . . . . . . . . . . . . . . . . . 23 (2 + 1) = 3
3432, 33syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (#‘⟨“𝐴𝐵𝐶”⟩) = 3)
3534oveq2d 6565 . . . . . . . . . . . . . . . . . . . . 21 ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = (0..^3))
36 fzo0to3tp 12421 . . . . . . . . . . . . . . . . . . . . 21 (0..^3) = {0, 1, 2}
3735, 36syl6eq 2660 . . . . . . . . . . . . . . . . . . . 20 ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = {0, 1, 2})
3831, 37syl5eleqr 2695 . . . . . . . . . . . . . . . . . . 19 ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → 1 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩)))
39 wrdsymbcl 13173 . . . . . . . . . . . . . . . . . . . 20 ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ 1 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩))) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ (Vtx‘𝐺))
4039, 2syl6eleqr 2699 . . . . . . . . . . . . . . . . . . 19 ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ 1 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩))) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ 𝑉)
4138, 40sylan2 490 . . . . . . . . . . . . . . . . . 18 ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ 𝑉)
4241adantr 480 . . . . . . . . . . . . . . . . 17 (((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) ∧ 𝐵𝑈) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ 𝑉)
4329, 42eqeltrd 2688 . . . . . . . . . . . . . . . 16 (((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) ∧ 𝐵𝑈) → 𝐵𝑉)
4443ex 449 . . . . . . . . . . . . . . 15 ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) → (𝐵𝑈𝐵𝑉))
4526, 44syl 17 . . . . . . . . . . . . . 14 (⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalkSN 𝐺) → (𝐵𝑈𝐵𝑉))
4645adantr 480 . . . . . . . . . . . . 13 ((⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalkSN 𝐺) ∧ ((⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)) → (𝐵𝑈𝐵𝑉))
4746a1i 11 . . . . . . . . . . . 12 ((𝐴𝑉𝐶𝑉) → ((⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalkSN 𝐺) ∧ ((⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)) → (𝐵𝑈𝐵𝑉)))
4825, 47syl5bi 231 . . . . . . . . . . 11 ((𝐴𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)} → (𝐵𝑈𝐵𝑉)))
4919, 48sylbid 229 . . . . . . . . . 10 ((𝐴𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐵𝑈𝐵𝑉)))
5049impd 446 . . . . . . . . 9 ((𝐴𝑉𝐶𝑉) → ((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵𝑈) → 𝐵𝑉))
5150impcom 445 . . . . . . . 8 (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵𝑈) ∧ (𝐴𝑉𝐶𝑉)) → 𝐵𝑉)
52 simprr 792 . . . . . . . 8 (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵𝑈) ∧ (𝐴𝑉𝐶𝑉)) → 𝐶𝑉)
539, 51, 523jca 1235 . . . . . . 7 (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵𝑈) ∧ (𝐴𝑉𝐶𝑉)) → (𝐴𝑉𝐵𝑉𝐶𝑉))
5453exp31 628 . . . . . 6 (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐵𝑈 → ((𝐴𝑉𝐶𝑉) → (𝐴𝑉𝐵𝑉𝐶𝑉))))
558, 54mpid 43 . . . . 5 (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐵𝑈 → (𝐴𝑉𝐵𝑉𝐶𝑉)))
566, 55syl6bi 242 . . . 4 (𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐵𝑈 → (𝐴𝑉𝐵𝑉𝐶𝑉))))
5756com23 84 . . 3 (𝐺 ∈ V → (𝐵𝑈 → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴𝑉𝐵𝑉𝐶𝑉))))
5857impd 446 . 2 (𝐺 ∈ V → ((𝐵𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴𝑉𝐵𝑉𝐶𝑉)))
59 df-wwlksnon 41035 . . . . . . . . . 10 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))
6059reldmmpt2 6669 . . . . . . . . 9 Rel dom WWalksNOn
6160ovprc2 6583 . . . . . . . 8 𝐺 ∈ V → (2 WWalksNOn 𝐺) = ∅)
6261oveqd 6566 . . . . . . 7 𝐺 ∈ V → (𝐴(2 WWalksNOn 𝐺)𝐶) = (𝐴𝐶))
63 0ov 6580 . . . . . . 7 (𝐴𝐶) = ∅
6462, 63syl6eq 2660 . . . . . 6 𝐺 ∈ V → (𝐴(2 WWalksNOn 𝐺)𝐶) = ∅)
6564eleq2d 2673 . . . . 5 𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ ∅))
66 noel 3878 . . . . . 6 ¬ ⟨“𝐴𝐵𝐶”⟩ ∈ ∅
6766pm2.21i 115 . . . . 5 (⟨“𝐴𝐵𝐶”⟩ ∈ ∅ → (𝐵𝑈 → (𝐴𝑉𝐵𝑉𝐶𝑉)))
6865, 67syl6bi 242 . . . 4 𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐵𝑈 → (𝐴𝑉𝐵𝑉𝐶𝑉))))
6968com23 84 . . 3 𝐺 ∈ V → (𝐵𝑈 → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴𝑉𝐵𝑉𝐶𝑉))))
7069impd 446 . 2 𝐺 ∈ V → ((𝐵𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴𝑉𝐵𝑉𝐶𝑉)))
7158, 70pm2.61i 175 1 ((𝐵𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴𝑉𝐵𝑉𝐶𝑉))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ∅c0 3874  {ctp 4129  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  3c3 10948  ℕ0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146  ⟨“cs3 13438  Vtxcvtx 25673   WWalkSN cwwlksn 41029   WWalksNOn cwwlksnon 41030 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-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-oadd 7451  df-er 7629  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-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-wwlks 41033  df-wwlksn 41034  df-wwlksnon 41035 This theorem is referenced by:  frgr2wwlkeqm  41496
 Copyright terms: Public domain W3C validator