MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wwlks2onv Structured version   Visualization version   GIF version

Theorem wwlks2onv 26716
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 11253 . . . . . . . 8 2 ∈ ℕ0
2 wwlks2onv.v . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
32wwlksnon 26607 . . . . . . . 8 ((2 ∈ ℕ0𝐺 ∈ V) → (2 WWalksNOn 𝐺) = (𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)}))
41, 3mpan 705 . . . . . . 7 (𝐺 ∈ V → (2 WWalksNOn 𝐺) = (𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)}))
54oveqd 6621 . . . . . 6 (𝐺 ∈ V → (𝐴(2 WWalksNOn 𝐺)𝐶) = (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶))
65eleq2d 2684 . . . . 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 2621 . . . . . . 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 6829 . . . . . 6 (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐴𝑉𝐶𝑉))
9 simprl 793 . . . . . . . 8 (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵𝑈) ∧ (𝐴𝑉𝐶𝑉)) → 𝐴𝑉)
10 eqeq2 2632 . . . . . . . . . . . . . . 15 (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴))
1110anbi1d 740 . . . . . . . . . . . . . 14 (𝑎 = 𝐴 → (((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝑐)))
1211rabbidv 3177 . . . . . . . . . . . . 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 2632 . . . . . . . . . . . . . . 15 (𝑐 = 𝐶 → ((𝑤‘2) = 𝑐 ↔ (𝑤‘2) = 𝐶))
1413anbi2d 739 . . . . . . . . . . . . . 14 (𝑐 = 𝐶 → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝑐) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)))
1514rabbidv 3177 . . . . . . . . . . . . 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 6632 . . . . . . . . . . . . . 14 (2 WWalksN 𝐺) ∈ V
1716rabex 4773 . . . . . . . . . . . . 13 {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)} ∈ V
1812, 15, 7, 17ovmpt2 6749 . . . . . . . . . . . 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 2684 . . . . . . . . . . 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 6147 . . . . . . . . . . . . . . 15 (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → (𝑤‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
2120eqeq1d 2623 . . . . . . . . . . . . . 14 (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → ((𝑤‘0) = 𝐴 ↔ (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴))
22 fveq1 6147 . . . . . . . . . . . . . . 15 (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → (𝑤‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
2322eqeq1d 2623 . . . . . . . . . . . . . 14 (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → ((𝑤‘2) = 𝐶 ↔ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶))
2421, 23anbi12d 746 . . . . . . . . . . . . 13 (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶) ↔ ((⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)))
2524elrab 3346 . . . . . . . . . . . 12 (⟨“𝐴𝐵𝐶”⟩ ∈ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)} ↔ (⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalksN 𝐺) ∧ ((⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)))
26 wwlknbp2 26621 . . . . . . . . . . . . . . 15 (⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalksN 𝐺) → (⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)))
27 s3fv1 13573 . . . . . . . . . . . . . . . . . . 19 (𝐵𝑈 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
2827eqcomd 2627 . . . . . . . . . . . . . . . . . 18 (𝐵𝑈𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1))
2928adantl 482 . . . . . . . . . . . . . . . . 17 (((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) ∧ 𝐵𝑈) → 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1))
30 1ex 9979 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ V
3130tpid2 4274 . . . . . . . . . . . . . . . . . . . 20 1 ∈ {0, 1, 2}
32 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1))
33 2p1e3 11095 . . . . . . . . . . . . . . . . . . . . . . 23 (2 + 1) = 3
3432, 33syl6eq 2671 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (#‘⟨“𝐴𝐵𝐶”⟩) = 3)
3534oveq2d 6620 . . . . . . . . . . . . . . . . . . . . 21 ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = (0..^3))
36 fzo0to3tp 12495 . . . . . . . . . . . . . . . . . . . . 21 (0..^3) = {0, 1, 2}
3735, 36syl6eq 2671 . . . . . . . . . . . . . . . . . . . 20 ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = {0, 1, 2})
3831, 37syl5eleqr 2705 . . . . . . . . . . . . . . . . . . 19 ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → 1 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩)))
39 wrdsymbcl 13257 . . . . . . . . . . . . . . . . . . . 20 ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ 1 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩))) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ (Vtx‘𝐺))
4039, 2syl6eleqr 2709 . . . . . . . . . . . . . . . . . . 19 ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ 1 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩))) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ 𝑉)
4138, 40sylan2 491 . . . . . . . . . . . . . . . . . 18 ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ 𝑉)
4241adantr 481 . . . . . . . . . . . . . . . . 17 (((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) ∧ 𝐵𝑈) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ 𝑉)
4329, 42eqeltrd 2698 . . . . . . . . . . . . . . . 16 (((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) ∧ 𝐵𝑈) → 𝐵𝑉)
4443ex 450 . . . . . . . . . . . . . . 15 ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) → (𝐵𝑈𝐵𝑉))
4526, 44syl 17 . . . . . . . . . . . . . 14 (⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalksN 𝐺) → (𝐵𝑈𝐵𝑉))
4645adantr 481 . . . . . . . . . . . . 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 232 . . . . . . . . . . 11 ((𝐴𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)} → (𝐵𝑈𝐵𝑉)))
4919, 48sylbid 230 . . . . . . . . . 10 ((𝐴𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐵𝑈𝐵𝑉)))
5049impd 447 . . . . . . . . 9 ((𝐴𝑉𝐶𝑉) → ((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵𝑈) → 𝐵𝑉))
5150impcom 446 . . . . . . . 8 (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵𝑈) ∧ (𝐴𝑉𝐶𝑉)) → 𝐵𝑉)
52 simprr 795 . . . . . . . 8 (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵𝑈) ∧ (𝐴𝑉𝐶𝑉)) → 𝐶𝑉)
539, 51, 523jca 1240 . . . . . . 7 (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵𝑈) ∧ (𝐴𝑉𝐶𝑉)) → (𝐴𝑉𝐵𝑉𝐶𝑉))
5453exp31 629 . . . . . 6 (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐵𝑈 → ((𝐴𝑉𝐶𝑉) → (𝐴𝑉𝐵𝑉𝐶𝑉))))
558, 54mpid 44 . . . . 5 (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎𝑉, 𝑐𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐵𝑈 → (𝐴𝑉𝐵𝑉𝐶𝑉)))
566, 55syl6bi 243 . . . 4 (𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐵𝑈 → (𝐴𝑉𝐵𝑉𝐶𝑉))))
5756com23 86 . . 3 (𝐺 ∈ V → (𝐵𝑈 → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴𝑉𝐵𝑉𝐶𝑉))))
5857impd 447 . 2 (𝐺 ∈ V → ((𝐵𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴𝑉𝐵𝑉𝐶𝑉)))
59 df-wwlksnon 26593 . . . . . . . . . 10 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))
6059reldmmpt2 6724 . . . . . . . . 9 Rel dom WWalksNOn
6160ovprc2 6638 . . . . . . . 8 𝐺 ∈ V → (2 WWalksNOn 𝐺) = ∅)
6261oveqd 6621 . . . . . . 7 𝐺 ∈ V → (𝐴(2 WWalksNOn 𝐺)𝐶) = (𝐴𝐶))
63 0ov 6635 . . . . . . 7 (𝐴𝐶) = ∅
6462, 63syl6eq 2671 . . . . . 6 𝐺 ∈ V → (𝐴(2 WWalksNOn 𝐺)𝐶) = ∅)
6564eleq2d 2684 . . . . 5 𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ ∅))
66 noel 3895 . . . . . 6 ¬ ⟨“𝐴𝐵𝐶”⟩ ∈ ∅
6766pm2.21i 116 . . . . 5 (⟨“𝐴𝐵𝐶”⟩ ∈ ∅ → (𝐵𝑈 → (𝐴𝑉𝐵𝑉𝐶𝑉)))
6865, 67syl6bi 243 . . . 4 𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐵𝑈 → (𝐴𝑉𝐵𝑉𝐶𝑉))))
6968com23 86 . . 3 𝐺 ∈ V → (𝐵𝑈 → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴𝑉𝐵𝑉𝐶𝑉))))
7069impd 447 . 2 𝐺 ∈ V → ((𝐵𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴𝑉𝐵𝑉𝐶𝑉)))
7158, 70pm2.61i 176 1 ((𝐵𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴𝑉𝐵𝑉𝐶𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  c0 3891  {ctp 4152  cfv 5847  (class class class)co 6604  cmpt2 6606  0cc0 9880  1c1 9881   + caddc 9883  2c2 11014  3c3 11015  0cn0 11236  ..^cfzo 12406  #chash 13057  Word cword 13230  ⟨“cs3 13524  Vtxcvtx 25774   WWalksN cwwlksn 26587   WWalksNOn cwwlksnon 26588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-s2 13530  df-s3 13531  df-wwlks 26591  df-wwlksn 26592  df-wwlksnon 26593
This theorem is referenced by:  frgr2wwlkeqm  27054
  Copyright terms: Public domain W3C validator