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Theorem 1pthon2v 29194
Description: For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
Hypotheses
Ref Expression
1pthon2v.v 𝑉 = (Vtxβ€˜πΊ)
1pthon2v.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
1pthon2v ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒) β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝)
Distinct variable groups:   𝐴,𝑒,𝑓,𝑝   𝐡,𝑒,𝑓,𝑝   𝑒,𝐺,𝑓,𝑝   𝑒,𝑉
Allowed substitution hints:   𝐸(𝑒,𝑓,𝑝)   𝑉(𝑓,𝑝)
Proof of Theorem 1pthon2v
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 simpl 483 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ 𝑉)
21anim2i 617 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐺 ∈ UHGraph ∧ 𝐴 ∈ 𝑉))
323adant3 1132 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒) β†’ (𝐺 ∈ UHGraph ∧ 𝐴 ∈ 𝑉))
43adantl 482 . . . . 5 ((𝐴 = 𝐡 ∧ (𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒)) β†’ (𝐺 ∈ UHGraph ∧ 𝐴 ∈ 𝑉))
5 1pthon2v.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
650pthonv 29170 . . . . 5 (𝐴 ∈ 𝑉 β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐴)𝑝)
74, 6simpl2im 504 . . . 4 ((𝐴 = 𝐡 ∧ (𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒)) β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐴)𝑝)
8 oveq2 7385 . . . . . . . 8 (𝐡 = 𝐴 β†’ (𝐴(PathsOnβ€˜πΊ)𝐡) = (𝐴(PathsOnβ€˜πΊ)𝐴))
98eqcoms 2739 . . . . . . 7 (𝐴 = 𝐡 β†’ (𝐴(PathsOnβ€˜πΊ)𝐡) = (𝐴(PathsOnβ€˜πΊ)𝐴))
109breqd 5136 . . . . . 6 (𝐴 = 𝐡 β†’ (𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝 ↔ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐴)𝑝))
11102exbidv 1927 . . . . 5 (𝐴 = 𝐡 β†’ (βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝 ↔ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐴)𝑝))
1211adantr 481 . . . 4 ((𝐴 = 𝐡 ∧ (𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒)) β†’ (βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝 ↔ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐴)𝑝))
137, 12mpbird 256 . . 3 ((𝐴 = 𝐡 ∧ (𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒)) β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝)
1413ex 413 . 2 (𝐴 = 𝐡 β†’ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒) β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝))
15 1pthon2v.e . . . . . . . . . . 11 𝐸 = (Edgβ€˜πΊ)
1615eleq2i 2824 . . . . . . . . . 10 (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edgβ€˜πΊ))
17 eqid 2731 . . . . . . . . . . 11 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
1817uhgredgiedgb 28174 . . . . . . . . . 10 (𝐺 ∈ UHGraph β†’ (𝑒 ∈ (Edgβ€˜πΊ) ↔ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ)𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)))
1916, 18bitrid 282 . . . . . . . . 9 (𝐺 ∈ UHGraph β†’ (𝑒 ∈ 𝐸 ↔ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ)𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)))
20193ad2ant1 1133 . . . . . . . 8 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) β†’ (𝑒 ∈ 𝐸 ↔ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ)𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)))
21 s1cli 14520 . . . . . . . . . . . 12 βŸ¨β€œπ‘–β€βŸ© ∈ Word V
22 s2cli 14796 . . . . . . . . . . . 12 βŸ¨β€œπ΄π΅β€βŸ© ∈ Word V
2321, 22pm3.2i 471 . . . . . . . . . . 11 (βŸ¨β€œπ‘–β€βŸ© ∈ Word V ∧ βŸ¨β€œπ΄π΅β€βŸ© ∈ Word V)
24 eqid 2731 . . . . . . . . . . . 12 βŸ¨β€œπ΄π΅β€βŸ© = βŸ¨β€œπ΄π΅β€βŸ©
25 eqid 2731 . . . . . . . . . . . 12 βŸ¨β€œπ‘–β€βŸ© = βŸ¨β€œπ‘–β€βŸ©
26 simpl2l 1226 . . . . . . . . . . . 12 (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) ∧ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ 𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)) ∧ {𝐴, 𝐡} βŠ† 𝑒)) β†’ 𝐴 ∈ 𝑉)
27 simpl2r 1227 . . . . . . . . . . . 12 (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) ∧ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ 𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)) ∧ {𝐴, 𝐡} βŠ† 𝑒)) β†’ 𝐡 ∈ 𝑉)
28 eqneqall 2950 . . . . . . . . . . . . . . . 16 (𝐴 = 𝐡 β†’ (𝐴 β‰  𝐡 β†’ ((iEdgβ€˜πΊ)β€˜π‘–) = {𝐴}))
2928com12 32 . . . . . . . . . . . . . . 15 (𝐴 β‰  𝐡 β†’ (𝐴 = 𝐡 β†’ ((iEdgβ€˜πΊ)β€˜π‘–) = {𝐴}))
30293ad2ant3 1135 . . . . . . . . . . . . . 14 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) β†’ (𝐴 = 𝐡 β†’ ((iEdgβ€˜πΊ)β€˜π‘–) = {𝐴}))
3130adantr 481 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) ∧ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ 𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)) ∧ {𝐴, 𝐡} βŠ† 𝑒)) β†’ (𝐴 = 𝐡 β†’ ((iEdgβ€˜πΊ)β€˜π‘–) = {𝐴}))
3231imp 407 . . . . . . . . . . . 12 ((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) ∧ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ 𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)) ∧ {𝐴, 𝐡} βŠ† 𝑒)) ∧ 𝐴 = 𝐡) β†’ ((iEdgβ€˜πΊ)β€˜π‘–) = {𝐴})
33 sseq2 3988 . . . . . . . . . . . . . . . 16 (𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ ({𝐴, 𝐡} βŠ† 𝑒 ↔ {𝐴, 𝐡} βŠ† ((iEdgβ€˜πΊ)β€˜π‘–)))
3433adantl 482 . . . . . . . . . . . . . . 15 ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ 𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ ({𝐴, 𝐡} βŠ† 𝑒 ↔ {𝐴, 𝐡} βŠ† ((iEdgβ€˜πΊ)β€˜π‘–)))
3534biimpa 477 . . . . . . . . . . . . . 14 (((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ 𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)) ∧ {𝐴, 𝐡} βŠ† 𝑒) β†’ {𝐴, 𝐡} βŠ† ((iEdgβ€˜πΊ)β€˜π‘–))
3635adantl 482 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) ∧ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ 𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)) ∧ {𝐴, 𝐡} βŠ† 𝑒)) β†’ {𝐴, 𝐡} βŠ† ((iEdgβ€˜πΊ)β€˜π‘–))
3736adantr 481 . . . . . . . . . . . 12 ((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) ∧ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ 𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)) ∧ {𝐴, 𝐡} βŠ† 𝑒)) ∧ 𝐴 β‰  𝐡) β†’ {𝐴, 𝐡} βŠ† ((iEdgβ€˜πΊ)β€˜π‘–))
3824, 25, 26, 27, 32, 37, 5, 171pthond 29185 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) ∧ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ 𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)) ∧ {𝐴, 𝐡} βŠ† 𝑒)) β†’ βŸ¨β€œπ‘–β€βŸ©(𝐴(PathsOnβ€˜πΊ)𝐡)βŸ¨β€œπ΄π΅β€βŸ©)
39 breq12 5130 . . . . . . . . . . . 12 ((𝑓 = βŸ¨β€œπ‘–β€βŸ© ∧ 𝑝 = βŸ¨β€œπ΄π΅β€βŸ©) β†’ (𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝 ↔ βŸ¨β€œπ‘–β€βŸ©(𝐴(PathsOnβ€˜πΊ)𝐡)βŸ¨β€œπ΄π΅β€βŸ©))
4039spc2egv 3572 . . . . . . . . . . 11 ((βŸ¨β€œπ‘–β€βŸ© ∈ Word V ∧ βŸ¨β€œπ΄π΅β€βŸ© ∈ Word V) β†’ (βŸ¨β€œπ‘–β€βŸ©(𝐴(PathsOnβ€˜πΊ)𝐡)βŸ¨β€œπ΄π΅β€βŸ© β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝))
4123, 38, 40mpsyl 68 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) ∧ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ 𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–)) ∧ {𝐴, 𝐡} βŠ† 𝑒)) β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝)
4241exp44 438 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) β†’ (𝑖 ∈ dom (iEdgβ€˜πΊ) β†’ (𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ ({𝐴, 𝐡} βŠ† 𝑒 β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝))))
4342rexlimdv 3152 . . . . . . . 8 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) β†’ (βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ)𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ ({𝐴, 𝐡} βŠ† 𝑒 β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝)))
4420, 43sylbid 239 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) β†’ (𝑒 ∈ 𝐸 β†’ ({𝐴, 𝐡} βŠ† 𝑒 β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝)))
4544rexlimdv 3152 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) β†’ (βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒 β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝))
46453exp 1119 . . . . 5 (𝐺 ∈ UHGraph β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴 β‰  𝐡 β†’ (βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒 β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝))))
4746com34 91 . . . 4 (𝐺 ∈ UHGraph β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒 β†’ (𝐴 β‰  𝐡 β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝))))
48473imp 1111 . . 3 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒) β†’ (𝐴 β‰  𝐡 β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝))
4948com12 32 . 2 (𝐴 β‰  𝐡 β†’ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒) β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝))
5014, 49pm2.61ine 3024 1 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒) β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2939  βˆƒwrex 3069  Vcvv 3459   βŠ† wss 3928  {csn 4606  {cpr 4608   class class class wbr 5125  dom cdm 5653  β€˜cfv 6516  (class class class)co 7377  Word cword 14429  βŸ¨β€œcs1 14510  βŸ¨β€œcs2 14757  Vtxcvtx 28044  iEdgciedg 28045  Edgcedg 28095  UHGraphcuhgr 28104  PathsOncpthson 28759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-int 4928  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-1st 7941  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8670  df-map 8789  df-pm 8790  df-en 8906  df-dom 8907  df-sdom 8908  df-fin 8909  df-card 9899  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-nn 12178  df-2 12240  df-n0 12438  df-z 12524  df-uz 12788  df-fz 13450  df-fzo 13593  df-hash 14256  df-word 14430  df-concat 14486  df-s1 14511  df-s2 14764  df-edg 28096  df-uhgr 28106  df-wlks 28644  df-wlkson 28645  df-trls 28737  df-trlson 28738  df-pths 28761  df-pthson 28763
This theorem is referenced by:  1pthon2ve  29195  cusconngr  29232
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