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Theorem umgr2wlk 41156
 Description: In a multigraph, there is a 1-walk of length 2 for each pair of adjacent edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)
Hypothesis
Ref Expression
umgr2wlk.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
umgr2wlk ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
Distinct variable groups:   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝   𝐶,𝑓,𝑝   𝑓,𝐺,𝑝
Allowed substitution hints:   𝐸(𝑓,𝑝)

Proof of Theorem umgr2wlk
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 umgruhgr 25770 . . . . . 6 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )
2 umgr2wlk.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
32eleq2i 2680 . . . . . . 7 ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ (Edg‘𝐺))
4 eqid 2610 . . . . . . . 8 (iEdg‘𝐺) = (iEdg‘𝐺)
54uhgredgiedgb 25799 . . . . . . 7 (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ (Edg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
63, 5syl5bb 271 . . . . . 6 (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
71, 6syl 17 . . . . 5 (𝐺 ∈ UMGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
87biimpd 218 . . . 4 (𝐺 ∈ UMGraph → ({𝐵, 𝐶} ∈ 𝐸 → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
98a1d 25 . . 3 (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 → ({𝐵, 𝐶} ∈ 𝐸 → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))))
1093imp 1249 . 2 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))
112eleq2i 2680 . . . . . . 7 ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ (Edg‘𝐺))
124uhgredgiedgb 25799 . . . . . . 7 (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ (Edg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
1311, 12syl5bb 271 . . . . . 6 (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
141, 13syl 17 . . . . 5 (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
1514biimpd 218 . . . 4 (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
1615a1dd 48 . . 3 (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 → ({𝐵, 𝐶} ∈ 𝐸 → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))))
17163imp 1249 . 2 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))
18 s2cli 13475 . . . . . . . . . 10 ⟨“𝑗𝑖”⟩ ∈ Word V
19 s3cli 13476 . . . . . . . . . 10 ⟨“𝐴𝐵𝐶”⟩ ∈ Word V
2018, 19pm3.2i 470 . . . . . . . . 9 (⟨“𝑗𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V)
21 eqid 2610 . . . . . . . . . 10 ⟨“𝑗𝑖”⟩ = ⟨“𝑗𝑖”⟩
22 eqid 2610 . . . . . . . . . 10 ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩
23 simpl1 1057 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → 𝐺 ∈ UMGraph )
24 3simpc 1053 . . . . . . . . . . 11 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
2524adantr 480 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
26 simpl 472 . . . . . . . . . . . 12 (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))
2726eqcomd 2616 . . . . . . . . . . 11 (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵})
2827adantl 481 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵})
29 simpr 476 . . . . . . . . . . . 12 (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))
3029eqcomd 2616 . . . . . . . . . . 11 (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐺)‘𝑖) = {𝐵, 𝐶})
3130adantl 481 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → ((iEdg‘𝐺)‘𝑖) = {𝐵, 𝐶})
322, 4, 21, 22, 23, 25, 28, 31umgr2adedgwlk 41152 . . . . . . . . 9 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → (⟨“𝑗𝑖”⟩(1Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (#‘⟨“𝑗𝑖”⟩) = 2 ∧ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))))
33 breq12 4588 . . . . . . . . . . 11 ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → (𝑓(1Walks‘𝐺)𝑝 ↔ ⟨“𝑗𝑖”⟩(1Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
34 fveq2 6103 . . . . . . . . . . . . 13 (𝑓 = ⟨“𝑗𝑖”⟩ → (#‘𝑓) = (#‘⟨“𝑗𝑖”⟩))
3534eqeq1d 2612 . . . . . . . . . . . 12 (𝑓 = ⟨“𝑗𝑖”⟩ → ((#‘𝑓) = 2 ↔ (#‘⟨“𝑗𝑖”⟩) = 2))
3635adantr 480 . . . . . . . . . . 11 ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((#‘𝑓) = 2 ↔ (#‘⟨“𝑗𝑖”⟩) = 2))
37 fveq1 6102 . . . . . . . . . . . . . 14 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
3837eqeq2d 2620 . . . . . . . . . . . . 13 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐴 = (𝑝‘0) ↔ 𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0)))
39 fveq1 6102 . . . . . . . . . . . . . 14 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘1) = (⟨“𝐴𝐵𝐶”⟩‘1))
4039eqeq2d 2620 . . . . . . . . . . . . 13 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐵 = (𝑝‘1) ↔ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1)))
41 fveq1 6102 . . . . . . . . . . . . . 14 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
4241eqeq2d 2620 . . . . . . . . . . . . 13 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐶 = (𝑝‘2) ↔ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2)))
4338, 40, 423anbi123d 1391 . . . . . . . . . . . 12 (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) ↔ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))))
4443adantl 481 . . . . . . . . . . 11 ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) ↔ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))))
4533, 36, 443anbi123d 1391 . . . . . . . . . 10 ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ (⟨“𝑗𝑖”⟩(1Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (#‘⟨“𝑗𝑖”⟩) = 2 ∧ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2)))))
4645spc2egv 3268 . . . . . . . . 9 ((⟨“𝑗𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V) → ((⟨“𝑗𝑖”⟩(1Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (#‘⟨“𝑗𝑖”⟩) = 2 ∧ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))) → ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
4720, 32, 46mpsyl 66 . . . . . . . 8 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
4847exp32 629 . . . . . . 7 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
4948com12 32 . . . . . 6 ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
5049rexlimivw 3011 . . . . 5 (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
5150com13 86 . . . 4 ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
5251rexlimivw 3011 . . 3 (∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
5352com12 32 . 2 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → (∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
5410, 17, 53mp2d 47 1 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  {cpr 4127   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  0cc0 9815  1c1 9816  2c2 10947  #chash 12979  Word cword 13146  ⟨“cs2 13437  ⟨“cs3 13438  iEdgciedg 25674   UHGraph cuhgr 25722   UMGraph cumgr 25748  Edgcedga 25792  1Walksc1wlks 40796 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-ifp 1007  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-rmo 2904  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-cda 8873  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-uhgr 25724  df-upgr 25749  df-umgr 25750  df-edga 25793  df-1wlks 40800 This theorem is referenced by:  umgr2wlkon  41157  umgrwwlks2on  41161
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