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Theorem wlkiswwlksur 26247
 Description: Lemma 3 for wlkiswwlkbij2 26249. (Contributed by Alexander van der Vekens, 23-Jul-2018.)
Hypotheses
Ref Expression
wlkiswwlkbij.t 𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}
wlkiswwlkbij.w 𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}
wlkiswwlkbij.f 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
Assertion
Ref Expression
wlkiswwlksur ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)
Distinct variable groups:   𝐸,𝑝,𝑡,𝑤   𝑁,𝑝,𝑡,𝑤   𝑃,𝑝,𝑡,𝑤   𝑡,𝑇   𝑉,𝑝,𝑡,𝑤   𝑡,𝑊   𝑤,𝐹   𝑤,𝑇   𝐹,𝑝   𝑇,𝑝   𝑊,𝑝
Allowed substitution hints:   𝐹(𝑡)   𝑊(𝑤)

Proof of Theorem wlkiswwlksur
Dummy variables 𝑓 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wlkiswwlkbij.t . . . 4 𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}
2 wlkiswwlkbij.w . . . 4 𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}
3 wlkiswwlkbij.f . . . 4 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
41, 2, 3wlkiswwlkfun 26245 . . 3 ((𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
543adant1 1072 . 2 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
6 fveq1 6102 . . . . . . 7 (𝑤 = 𝑝 → (𝑤‘0) = (𝑝‘0))
76eqeq1d 2612 . . . . . 6 (𝑤 = 𝑝 → ((𝑤‘0) = 𝑃 ↔ (𝑝‘0) = 𝑃))
87, 2elrab2 3333 . . . . 5 (𝑝𝑊 ↔ (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑝‘0) = 𝑃))
9 wlklniswwlkn 26229 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → (∃𝑓(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 𝑁) ↔ 𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
10 df-br 4584 . . . . . . . . . . . . 13 (𝑓(𝑉 Walks 𝐸)𝑝 ↔ ⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸))
11 vex 3176 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
12 vex 3176 . . . . . . . . . . . . . . . . 17 𝑝 ∈ V
1311, 12op1st 7067 . . . . . . . . . . . . . . . 16 (1st ‘⟨𝑓, 𝑝⟩) = 𝑓
1413eqcomi 2619 . . . . . . . . . . . . . . 15 𝑓 = (1st ‘⟨𝑓, 𝑝⟩)
1514fveq2i 6106 . . . . . . . . . . . . . 14 (#‘𝑓) = (#‘(1st ‘⟨𝑓, 𝑝⟩))
1615eqeq1i 2615 . . . . . . . . . . . . 13 ((#‘𝑓) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
1711, 12op2nd 7068 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨𝑓, 𝑝⟩) = 𝑝
1817eqcomi 2619 . . . . . . . . . . . . . . . 16 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩)
1918fveq1i 6104 . . . . . . . . . . . . . . 15 (𝑝‘0) = ((2nd ‘⟨𝑓, 𝑝⟩)‘0)
2019eqeq1i 2615 . . . . . . . . . . . . . 14 ((𝑝‘0) = 𝑃 ↔ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)
21 opex 4859 . . . . . . . . . . . . . . . 16 𝑓, 𝑝⟩ ∈ V
2221a1i 11 . . . . . . . . . . . . . . 15 ((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ⟨𝑓, 𝑝⟩ ∈ V)
23 simpll 786 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → ⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸))
24 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
2524anim1i 590 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃))
2618a1i 11 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩))
2723, 25, 26jca31 555 . . . . . . . . . . . . . . . . 17 (((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → ((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩)))
28 eleq1 2676 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑢 ∈ (𝑉 Walks 𝐸) ↔ ⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸)))
29 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = ⟨𝑓, 𝑝⟩ → (1st𝑢) = (1st ‘⟨𝑓, 𝑝⟩))
3029fveq2d 6107 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = ⟨𝑓, 𝑝⟩ → (#‘(1st𝑢)) = (#‘(1st ‘⟨𝑓, 𝑝⟩)))
3130eqeq1d 2612 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = ⟨𝑓, 𝑝⟩ → ((#‘(1st𝑢)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
32 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = ⟨𝑓, 𝑝⟩ → (2nd𝑢) = (2nd ‘⟨𝑓, 𝑝⟩))
3332fveq1d 6105 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = ⟨𝑓, 𝑝⟩ → ((2nd𝑢)‘0) = ((2nd ‘⟨𝑓, 𝑝⟩)‘0))
3433eqeq1d 2612 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = ⟨𝑓, 𝑝⟩ → (((2nd𝑢)‘0) = 𝑃 ↔ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃))
3531, 34anbi12d 743 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃) ↔ ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)))
3628, 35anbi12d 743 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → ((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ↔ (⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃))))
3732eqeq2d 2620 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑝 = (2nd𝑢) ↔ 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩)))
3836, 37anbi12d 743 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨𝑓, 𝑝⟩ → (((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢)) ↔ ((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩))))
3927, 38syl5ibrcom 236 . . . . . . . . . . . . . . . 16 (((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → (𝑢 = ⟨𝑓, 𝑝⟩ → ((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
4039impancom 455 . . . . . . . . . . . . . . 15 (((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → (((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃 → ((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
4122, 40spcimedv 3265 . . . . . . . . . . . . . 14 ((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → (((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
4220, 41syl5bi 231 . . . . . . . . . . . . 13 ((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
4310, 16, 42syl2anb 495 . . . . . . . . . . . 12 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
4443exlimiv 1845 . . . . . . . . . . 11 (∃𝑓(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢))))
459, 44syl6bir 243 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢)))))
4645imp32 448 . . . . . . . . 9 ((𝑉 USGrph 𝐸 ∧ (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢)))
47 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑝 = 𝑢 → (1st𝑝) = (1st𝑢))
4847fveq2d 6107 . . . . . . . . . . . . . 14 (𝑝 = 𝑢 → (#‘(1st𝑝)) = (#‘(1st𝑢)))
4948eqeq1d 2612 . . . . . . . . . . . . 13 (𝑝 = 𝑢 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑢)) = 𝑁))
50 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑝 = 𝑢 → (2nd𝑝) = (2nd𝑢))
5150fveq1d 6105 . . . . . . . . . . . . . 14 (𝑝 = 𝑢 → ((2nd𝑝)‘0) = ((2nd𝑢)‘0))
5251eqeq1d 2612 . . . . . . . . . . . . 13 (𝑝 = 𝑢 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑢)‘0) = 𝑃))
5349, 52anbi12d 743 . . . . . . . . . . . 12 (𝑝 = 𝑢 → (((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)))
5453elrab 3331 . . . . . . . . . . 11 (𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)} ↔ (𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)))
5554anbi1i 727 . . . . . . . . . 10 ((𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd𝑢)) ↔ ((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢)))
5655exbii 1764 . . . . . . . . 9 (∃𝑢(𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd𝑢)) ↔ ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd𝑢)))
5746, 56sylibr 223 . . . . . . . 8 ((𝑉 USGrph 𝐸 ∧ (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢(𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd𝑢)))
58 df-rex 2902 . . . . . . . 8 (∃𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}𝑝 = (2nd𝑢) ↔ ∃𝑢(𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd𝑢)))
5957, 58sylibr 223 . . . . . . 7 ((𝑉 USGrph 𝐸 ∧ (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}𝑝 = (2nd𝑢))
601rexeqi 3120 . . . . . . 7 (∃𝑢𝑇 𝑝 = (2nd𝑢) ↔ ∃𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}𝑝 = (2nd𝑢))
6159, 60sylibr 223 . . . . . 6 ((𝑉 USGrph 𝐸 ∧ (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢𝑇 𝑝 = (2nd𝑢))
62 fveq2 6103 . . . . . . . . 9 (𝑡 = 𝑢 → (2nd𝑡) = (2nd𝑢))
63 fvex 6113 . . . . . . . . 9 (2nd𝑢) ∈ V
6462, 3, 63fvmpt 6191 . . . . . . . 8 (𝑢𝑇 → (𝐹𝑢) = (2nd𝑢))
6564eqeq2d 2620 . . . . . . 7 (𝑢𝑇 → (𝑝 = (𝐹𝑢) ↔ 𝑝 = (2nd𝑢)))
6665rexbiia 3022 . . . . . 6 (∃𝑢𝑇 𝑝 = (𝐹𝑢) ↔ ∃𝑢𝑇 𝑝 = (2nd𝑢))
6761, 66sylibr 223 . . . . 5 ((𝑉 USGrph 𝐸 ∧ (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
688, 67sylan2b 491 . . . 4 ((𝑉 USGrph 𝐸𝑝𝑊) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
6968ralrimiva 2949 . . 3 (𝑉 USGrph 𝐸 → ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢))
70693ad2ant1 1075 . 2 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢))
71 dffo3 6282 . 2 (𝐹:𝑇onto𝑊 ↔ (𝐹:𝑇𝑊 ∧ ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢)))
725, 70, 71sylanbrc 695 1 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  ℕ0cn0 11169  #chash 12979   USGrph cusg 25859   Walks cwalk 26026   WWalksN cwwlkn 26206 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-fal 1481  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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wwlk 26207  df-wwlkn 26208 This theorem is referenced by:  wlkiswwlkbij  26248
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