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Theorem wlkwwlksur 26998
Description: Lemma 3 for wlkwwlkbij2 27000. (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 26262 . . 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 26996 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
61, 5syl3an1 1166 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
7 fveq1 6343 . . . . . . 7 (𝑤 = 𝑝 → (𝑤‘0) = (𝑝‘0))
87eqeq1d 2754 . . . . . 6 (𝑤 = 𝑝 → ((𝑤‘0) = 𝑃 ↔ (𝑝‘0) = 𝑃))
98, 3elrab2 3499 . . . . 5 (𝑝𝑊 ↔ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃))
10 wlklnwwlkn 26985 . . . . . . . . . . 11 (𝐺 ∈ USPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑝 ∈ (𝑁 WWalksN 𝐺)))
11 df-br 4797 . . . . . . . . . . . . 13 (𝑓(Walks‘𝐺)𝑝 ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺))
12 vex 3335 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
13 vex 3335 . . . . . . . . . . . . . . . . 17 𝑝 ∈ V
1412, 13op1st 7333 . . . . . . . . . . . . . . . 16 (1st ‘⟨𝑓, 𝑝⟩) = 𝑓
1514eqcomi 2761 . . . . . . . . . . . . . . 15 𝑓 = (1st ‘⟨𝑓, 𝑝⟩)
1615fveq2i 6347 . . . . . . . . . . . . . 14 (♯‘𝑓) = (♯‘(1st ‘⟨𝑓, 𝑝⟩))
1716eqeq1i 2757 . . . . . . . . . . . . 13 ((♯‘𝑓) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
1812, 13op2nd 7334 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨𝑓, 𝑝⟩) = 𝑝
1918eqcomi 2761 . . . . . . . . . . . . . . . 16 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩)
2019fveq1i 6345 . . . . . . . . . . . . . . 15 (𝑝‘0) = ((2nd ‘⟨𝑓, 𝑝⟩)‘0)
2120eqeq1i 2757 . . . . . . . . . . . . . 14 ((𝑝‘0) = 𝑃 ↔ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)
22 opex 5073 . . . . . . . . . . . . . . . 16 𝑓, 𝑝⟩ ∈ V
2322a1i 11 . . . . . . . . . . . . . . 15 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ⟨𝑓, 𝑝⟩ ∈ V)
24 simpll 807 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺))
25 simpr 479 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
2625anim1i 593 . . . . . . . . . . . . . . . . . 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 558 . . . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑢 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺)))
30 fveq2 6344 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = ⟨𝑓, 𝑝⟩ → (1st𝑢) = (1st ‘⟨𝑓, 𝑝⟩))
3130fveq2d 6348 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = ⟨𝑓, 𝑝⟩ → (♯‘(1st𝑢)) = (♯‘(1st ‘⟨𝑓, 𝑝⟩)))
3231eqeq1d 2754 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = ⟨𝑓, 𝑝⟩ → ((♯‘(1st𝑢)) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
33 fveq2 6344 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = ⟨𝑓, 𝑝⟩ → (2nd𝑢) = (2nd ‘⟨𝑓, 𝑝⟩))
3433fveq1d 6346 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = ⟨𝑓, 𝑝⟩ → ((2nd𝑢)‘0) = ((2nd ‘⟨𝑓, 𝑝⟩)‘0))
3534eqeq1d 2754 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = ⟨𝑓, 𝑝⟩ → (((2nd𝑢)‘0) = 𝑃 ↔ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃))
3632, 35anbi12d 749 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (((♯‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃) ↔ ((♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)))
3729, 36anbi12d 749 . . . . . . . . . . . . . . . . . 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 2762 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑝 = (2nd𝑢) ↔ 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩)))
3937, 38anbi12d 749 . . . . . . . . . . . . . . . . 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 3424 . . . . . . . . . . . . . 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 497 . . . . . . . . . . . 12 ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
4544exlimiv 1999 . . . . . . . . . . 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 6344 . . . . . . . . . . . . . . 15 (𝑝 = 𝑢 → (1st𝑝) = (1st𝑢))
4948fveq2d 6348 . . . . . . . . . . . . . 14 (𝑝 = 𝑢 → (♯‘(1st𝑝)) = (♯‘(1st𝑢)))
5049eqeq1d 2754 . . . . . . . . . . . . 13 (𝑝 = 𝑢 → ((♯‘(1st𝑝)) = 𝑁 ↔ (♯‘(1st𝑢)) = 𝑁))
51 fveq2 6344 . . . . . . . . . . . . . . 15 (𝑝 = 𝑢 → (2nd𝑝) = (2nd𝑢))
5251fveq1d 6346 . . . . . . . . . . . . . 14 (𝑝 = 𝑢 → ((2nd𝑝)‘0) = ((2nd𝑢)‘0))
5352eqeq1d 2754 . . . . . . . . . . . . 13 (𝑝 = 𝑢 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑢)‘0) = 𝑃))
5450, 53anbi12d 749 . . . . . . . . . . . 12 (𝑝 = 𝑢 → (((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((♯‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)))
5554elrab 3496 . . . . . . . . . . 11 (𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)} ↔ (𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)))
5655anbi1i 733 . . . . . . . . . 10 ((𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd𝑢)) ↔ ((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢)))
5756exbii 1915 . . . . . . . . 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 3048 . . . . . . . 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 3274 . . . . . . 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 6344 . . . . . . . . 9 (𝑡 = 𝑢 → (2nd𝑡) = (2nd𝑢))
64 fvex 6354 . . . . . . . . 9 (2nd𝑢) ∈ V
6563, 4, 64fvmpt 6436 . . . . . . . 8 (𝑢𝑇 → (𝐹𝑢) = (2nd𝑢))
6665eqeq2d 2762 . . . . . . 7 (𝑢𝑇 → (𝑝 = (𝐹𝑢) ↔ 𝑝 = (2nd𝑢)))
6766rexbiia 3170 . . . . . 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 493 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑝𝑊) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
7069ralrimiva 3096 . . 3 (𝐺 ∈ USPGraph → ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢))
71703ad2ant1 1127 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢))
72 dffo3 6529 . 2 (𝐹:𝑇onto𝑊 ↔ (𝐹:𝑇𝑊 ∧ ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢)))
736, 71, 72sylanbrc 701 1 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1624  wex 1845  wcel 2131  wral 3042  wrex 3043  {crab 3046  Vcvv 3332  cop 4319   class class class wbr 4796  cmpt 4873  wf 6037  ontowfo 6039  cfv 6041  (class class class)co 6805  1st c1st 7323  2nd c2nd 7324  0cc0 10120  0cn0 11476  chash 13303  UPGraphcupgr 26166  USPGraphcuspgr 26234  Walkscwlks 26694   WWalksN cwwlksn 26921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-map 8017  df-pm 8018  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8947  df-cda 9174  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-n0 11477  df-xnn0 11548  df-z 11562  df-uz 11872  df-fz 12512  df-fzo 12652  df-hash 13304  df-word 13477  df-edg 26131  df-uhgr 26144  df-upgr 26168  df-uspgr 26236  df-wlks 26697  df-wwlks 26925  df-wwlksn 26926
This theorem is referenced by:  wlkwwlkbij  26999
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