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Theorem 1pthon2v 29673
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 481 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ 𝑉)
21anim2i 615 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐺 ∈ UHGraph ∧ 𝐴 ∈ 𝑉))
323adant3 1130 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒) β†’ (𝐺 ∈ UHGraph ∧ 𝐴 ∈ 𝑉))
43adantl 480 . . . . 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 29649 . . . . 5 (𝐴 ∈ 𝑉 β†’ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐴)𝑝)
74, 6simpl2im 502 . . . 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 7419 . . . . . . . 8 (𝐡 = 𝐴 β†’ (𝐴(PathsOnβ€˜πΊ)𝐡) = (𝐴(PathsOnβ€˜πΊ)𝐴))
98eqcoms 2738 . . . . . . 7 (𝐴 = 𝐡 β†’ (𝐴(PathsOnβ€˜πΊ)𝐡) = (𝐴(PathsOnβ€˜πΊ)𝐴))
109breqd 5158 . . . . . 6 (𝐴 = 𝐡 β†’ (𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝 ↔ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐴)𝑝))
11102exbidv 1925 . . . . 5 (𝐴 = 𝐡 β†’ (βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐡)𝑝 ↔ βˆƒπ‘“βˆƒπ‘ 𝑓(𝐴(PathsOnβ€˜πΊ)𝐴)𝑝))
1211adantr 479 . . . 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 411 . 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 2823 . . . . . . . . . 10 (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edgβ€˜πΊ))
17 eqid 2730 . . . . . . . . . . 11 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
1817uhgredgiedgb 28653 . . . . . . . . . 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 1131 . . . . . . . 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 14559 . . . . . . . . . . . 12 βŸ¨β€œπ‘–β€βŸ© ∈ Word V
22 s2cli 14835 . . . . . . . . . . . 12 βŸ¨β€œπ΄π΅β€βŸ© ∈ Word V
2321, 22pm3.2i 469 . . . . . . . . . . 11 (βŸ¨β€œπ‘–β€βŸ© ∈ Word V ∧ βŸ¨β€œπ΄π΅β€βŸ© ∈ Word V)
24 eqid 2730 . . . . . . . . . . . 12 βŸ¨β€œπ΄π΅β€βŸ© = βŸ¨β€œπ΄π΅β€βŸ©
25 eqid 2730 . . . . . . . . . . . 12 βŸ¨β€œπ‘–β€βŸ© = βŸ¨β€œπ‘–β€βŸ©
26 simpl2l 1224 . . . . . . . . . . . 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 1225 . . . . . . . . . . . 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 2949 . . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . . 14 ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) β†’ (𝐴 = 𝐡 β†’ ((iEdgβ€˜πΊ)β€˜π‘–) = {𝐴}))
3130adantr 479 . . . . . . . . . . . . 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 405 . . . . . . . . . . . 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 4007 . . . . . . . . . . . . . . . 16 (𝑒 = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ ({𝐴, 𝐡} βŠ† 𝑒 ↔ {𝐴, 𝐡} βŠ† ((iEdgβ€˜πΊ)β€˜π‘–)))
3433adantl 480 . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 29664 . . . . . . . . . . 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 5152 . . . . . . . . . . . 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 3588 . . . . . . . . . . 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 436 . . . . . . . . 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 3151 . . . . . . . 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 3151 . . . . . 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 1117 . . . . 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 1109 . . 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 3023 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 394   ∧ w3a 1085   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104   β‰  wne 2938  βˆƒwrex 3068  Vcvv 3472   βŠ† wss 3947  {csn 4627  {cpr 4629   class class class wbr 5147  dom cdm 5675  β€˜cfv 6542  (class class class)co 7411  Word cword 14468  βŸ¨β€œcs1 14549  βŸ¨β€œcs2 14796  Vtxcvtx 28523  iEdgciedg 28524  Edgcedg 28574  UHGraphcuhgr 28583  PathsOncpthson 29238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-concat 14525  df-s1 14550  df-s2 14803  df-edg 28575  df-uhgr 28585  df-wlks 29123  df-wlkson 29124  df-trls 29216  df-trlson 29217  df-pths 29240  df-pthson 29242
This theorem is referenced by:  1pthon2ve  29674  cusconngr  29711
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