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Theorem wlkwwlksur 26851
Description: Lemma 3 for wlkwwlkbij2 26853. (Contributed by Alexander van der Vekens, 23-Jul-2018.) (Revised by AV, 16-Apr-2021.)
Hypotheses
Ref Expression
wlkwwlkbij.t 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}
wlkwwlkbij.w 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
wlkwwlkbij.f 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
Assertion
Ref Expression
wlkwwlksur ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)
Distinct variable groups:   𝐺,𝑝,𝑡,𝑤   𝑁,𝑝,𝑡,𝑤   𝑃,𝑝,𝑡,𝑤   𝑡,𝑇,𝑤   𝑡,𝑉   𝑡,𝑊   𝑤,𝐹   𝑤,𝑉   𝐹,𝑝   𝑇,𝑝   𝑊,𝑝
Allowed substitution hints:   𝐹(𝑡)   𝑉(𝑝)   𝑊(𝑤)

Proof of Theorem wlkwwlksur
Dummy variables 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uspgrupgr 26116 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
2 wlkwwlkbij.t . . . 4 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}
3 wlkwwlkbij.w . . . 4 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
4 wlkwwlkbij.f . . . 4 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
52, 3, 4wlkwwlkfun 26849 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
61, 5syl3an1 1399 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
7 fveq1 6228 . . . . . . 7 (𝑤 = 𝑝 → (𝑤‘0) = (𝑝‘0))
87eqeq1d 2653 . . . . . 6 (𝑤 = 𝑝 → ((𝑤‘0) = 𝑃 ↔ (𝑝‘0) = 𝑃))
98, 3elrab2 3399 . . . . 5 (𝑝𝑊 ↔ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃))
10 wlklnwwlkn 26838 . . . . . . . . . . 11 (𝐺 ∈ USPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 𝑁) ↔ 𝑝 ∈ (𝑁 WWalksN 𝐺)))
11 df-br 4686 . . . . . . . . . . . . 13 (𝑓(Walks‘𝐺)𝑝 ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺))
12 vex 3234 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
13 vex 3234 . . . . . . . . . . . . . . . . 17 𝑝 ∈ V
1412, 13op1st 7218 . . . . . . . . . . . . . . . 16 (1st ‘⟨𝑓, 𝑝⟩) = 𝑓
1514eqcomi 2660 . . . . . . . . . . . . . . 15 𝑓 = (1st ‘⟨𝑓, 𝑝⟩)
1615fveq2i 6232 . . . . . . . . . . . . . 14 (#‘𝑓) = (#‘(1st ‘⟨𝑓, 𝑝⟩))
1716eqeq1i 2656 . . . . . . . . . . . . 13 ((#‘𝑓) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
1812, 13op2nd 7219 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨𝑓, 𝑝⟩) = 𝑝
1918eqcomi 2660 . . . . . . . . . . . . . . . 16 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩)
2019fveq1i 6230 . . . . . . . . . . . . . . 15 (𝑝‘0) = ((2nd ‘⟨𝑓, 𝑝⟩)‘0)
2120eqeq1i 2656 . . . . . . . . . . . . . 14 ((𝑝‘0) = 𝑃 ↔ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)
22 opex 4962 . . . . . . . . . . . . . . . 16 𝑓, 𝑝⟩ ∈ V
2322a1i 11 . . . . . . . . . . . . . . 15 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ⟨𝑓, 𝑝⟩ ∈ V)
24 simpll 805 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺))
25 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
2625anim1i 591 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃))
2719a1i 11 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩))
2824, 26, 27jca31 556 . . . . . . . . . . . . . . . . 17 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩)))
29 eleq1 2718 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑢 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺)))
30 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = ⟨𝑓, 𝑝⟩ → (1st𝑢) = (1st ‘⟨𝑓, 𝑝⟩))
3130fveq2d 6233 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = ⟨𝑓, 𝑝⟩ → (#‘(1st𝑢)) = (#‘(1st ‘⟨𝑓, 𝑝⟩)))
3231eqeq1d 2653 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = ⟨𝑓, 𝑝⟩ → ((#‘(1st𝑢)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
33 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = ⟨𝑓, 𝑝⟩ → (2nd𝑢) = (2nd ‘⟨𝑓, 𝑝⟩))
3433fveq1d 6231 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = ⟨𝑓, 𝑝⟩ → ((2nd𝑢)‘0) = ((2nd ‘⟨𝑓, 𝑝⟩)‘0))
3534eqeq1d 2653 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = ⟨𝑓, 𝑝⟩ → (((2nd𝑢)‘0) = 𝑃 ↔ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃))
3632, 35anbi12d 747 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃) ↔ ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)))
3729, 36anbi12d 747 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → ((𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ↔ (⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃))))
3833eqeq2d 2661 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑝 = (2nd𝑢) ↔ 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩)))
3937, 38anbi12d 747 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨𝑓, 𝑝⟩ → (((𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢)) ↔ ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩))))
4028, 39syl5ibrcom 237 . . . . . . . . . . . . . . . 16 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → (𝑢 = ⟨𝑓, 𝑝⟩ → ((𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
4140impancom 455 . . . . . . . . . . . . . . 15 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → (((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃 → ((𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
4223, 41spcimedv 3323 . . . . . . . . . . . . . 14 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → (((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
4321, 42syl5bi 232 . . . . . . . . . . . . 13 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
4411, 17, 43syl2anb 495 . . . . . . . . . . . 12 ((𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
4544exlimiv 1898 . . . . . . . . . . 11 (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
4610, 45syl6bir 244 . . . . . . . . . 10 (𝐺 ∈ USPGraph → (𝑝 ∈ (𝑁 WWalksN 𝐺) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢)))))
4746imp32 448 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢)))
48 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑝 = 𝑢 → (1st𝑝) = (1st𝑢))
4948fveq2d 6233 . . . . . . . . . . . . . 14 (𝑝 = 𝑢 → (#‘(1st𝑝)) = (#‘(1st𝑢)))
5049eqeq1d 2653 . . . . . . . . . . . . 13 (𝑝 = 𝑢 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑢)) = 𝑁))
51 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑝 = 𝑢 → (2nd𝑝) = (2nd𝑢))
5251fveq1d 6231 . . . . . . . . . . . . . 14 (𝑝 = 𝑢 → ((2nd𝑝)‘0) = ((2nd𝑢)‘0))
5352eqeq1d 2653 . . . . . . . . . . . . 13 (𝑝 = 𝑢 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑢)‘0) = 𝑃))
5450, 53anbi12d 747 . . . . . . . . . . . 12 (𝑝 = 𝑢 → (((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)))
5554elrab 3396 . . . . . . . . . . 11 (𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)} ↔ (𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)))
5655anbi1i 731 . . . . . . . . . 10 ((𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd𝑢)) ↔ ((𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢)))
5756exbii 1814 . . . . . . . . 9 (∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd𝑢)) ↔ ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢)))
5847, 57sylibr 224 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd𝑢)))
59 df-rex 2947 . . . . . . . 8 (∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}𝑝 = (2nd𝑢) ↔ ∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd𝑢)))
6058, 59sylibr 224 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}𝑝 = (2nd𝑢))
612rexeqi 3173 . . . . . . 7 (∃𝑢𝑇 𝑝 = (2nd𝑢) ↔ ∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}𝑝 = (2nd𝑢))
6260, 61sylibr 224 . . . . . 6 ((𝐺 ∈ USPGraph ∧ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢𝑇 𝑝 = (2nd𝑢))
63 fveq2 6229 . . . . . . . . 9 (𝑡 = 𝑢 → (2nd𝑡) = (2nd𝑢))
64 fvex 6239 . . . . . . . . 9 (2nd𝑢) ∈ V
6563, 4, 64fvmpt 6321 . . . . . . . 8 (𝑢𝑇 → (𝐹𝑢) = (2nd𝑢))
6665eqeq2d 2661 . . . . . . 7 (𝑢𝑇 → (𝑝 = (𝐹𝑢) ↔ 𝑝 = (2nd𝑢)))
6766rexbiia 3069 . . . . . 6 (∃𝑢𝑇 𝑝 = (𝐹𝑢) ↔ ∃𝑢𝑇 𝑝 = (2nd𝑢))
6862, 67sylibr 224 . . . . 5 ((𝐺 ∈ USPGraph ∧ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
699, 68sylan2b 491 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑝𝑊) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
7069ralrimiva 2995 . . 3 (𝐺 ∈ USPGraph → ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢))
71703ad2ant1 1102 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢))
72 dffo3 6414 . 2 (𝐹:𝑇onto𝑊 ↔ (𝐹:𝑇𝑊 ∧ ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢)))
736, 71, 72sylanbrc 699 1 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cop 4216   class class class wbr 4685  cmpt 4762  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  0cc0 9974  0cn0 11330  #chash 13157  UPGraphcupgr 26020  USPGraphcuspgr 26088  Walkscwlks 26548   WWalksN cwwlksn 26774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-edg 25985  df-uhgr 25998  df-upgr 26022  df-uspgr 26090  df-wlks 26551  df-wwlks 26778  df-wwlksn 26779
This theorem is referenced by:  wlkwwlkbij  26852
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