Metamath Proof Explorer


Theorem isuspgr

Description: The property of being a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017) (Revised by AV, 13-Oct-2020)

Ref Expression
Hypotheses isuspgr.v 𝑉 = ( Vtx ‘ 𝐺 )
isuspgr.e 𝐸 = ( iEdg ‘ 𝐺 )
Assertion isuspgr ( 𝐺𝑈 → ( 𝐺 ∈ USPGraph ↔ 𝐸 : dom 𝐸1-1→ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )

Proof

Step Hyp Ref Expression
1 isuspgr.v 𝑉 = ( Vtx ‘ 𝐺 )
2 isuspgr.e 𝐸 = ( iEdg ‘ 𝐺 )
3 df-uspgr USPGraph = { 𝑔[ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒1-1→ { 𝑥 ∈ ( 𝒫 𝑣 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } }
4 3 eleq2i ( 𝐺 ∈ USPGraph ↔ 𝐺 ∈ { 𝑔[ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒1-1→ { 𝑥 ∈ ( 𝒫 𝑣 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } } )
5 fveq2 ( = 𝐺 → ( iEdg ‘ ) = ( iEdg ‘ 𝐺 ) )
6 5 2 eqtr4di ( = 𝐺 → ( iEdg ‘ ) = 𝐸 )
7 5 dmeqd ( = 𝐺 → dom ( iEdg ‘ ) = dom ( iEdg ‘ 𝐺 ) )
8 2 eqcomi ( iEdg ‘ 𝐺 ) = 𝐸
9 8 dmeqi dom ( iEdg ‘ 𝐺 ) = dom 𝐸
10 7 9 eqtrdi ( = 𝐺 → dom ( iEdg ‘ ) = dom 𝐸 )
11 fveq2 ( = 𝐺 → ( Vtx ‘ ) = ( Vtx ‘ 𝐺 ) )
12 11 1 eqtr4di ( = 𝐺 → ( Vtx ‘ ) = 𝑉 )
13 12 pweqd ( = 𝐺 → 𝒫 ( Vtx ‘ ) = 𝒫 𝑉 )
14 13 difeq1d ( = 𝐺 → ( 𝒫 ( Vtx ‘ ) ∖ { ∅ } ) = ( 𝒫 𝑉 ∖ { ∅ } ) )
15 14 rabeqdv ( = 𝐺 → { 𝑥 ∈ ( 𝒫 ( Vtx ‘ ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } = { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
16 6 10 15 f1eq123d ( = 𝐺 → ( ( iEdg ‘ ) : dom ( iEdg ‘ ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ↔ 𝐸 : dom 𝐸1-1→ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
17 fvexd ( 𝑔 = → ( Vtx ‘ 𝑔 ) ∈ V )
18 fveq2 ( 𝑔 = → ( Vtx ‘ 𝑔 ) = ( Vtx ‘ ) )
19 fvexd ( ( 𝑔 = 𝑣 = ( Vtx ‘ ) ) → ( iEdg ‘ 𝑔 ) ∈ V )
20 fveq2 ( 𝑔 = → ( iEdg ‘ 𝑔 ) = ( iEdg ‘ ) )
21 20 adantr ( ( 𝑔 = 𝑣 = ( Vtx ‘ ) ) → ( iEdg ‘ 𝑔 ) = ( iEdg ‘ ) )
22 simpr ( ( ( 𝑔 = 𝑣 = ( Vtx ‘ ) ) ∧ 𝑒 = ( iEdg ‘ ) ) → 𝑒 = ( iEdg ‘ ) )
23 22 dmeqd ( ( ( 𝑔 = 𝑣 = ( Vtx ‘ ) ) ∧ 𝑒 = ( iEdg ‘ ) ) → dom 𝑒 = dom ( iEdg ‘ ) )
24 pweq ( 𝑣 = ( Vtx ‘ ) → 𝒫 𝑣 = 𝒫 ( Vtx ‘ ) )
25 24 ad2antlr ( ( ( 𝑔 = 𝑣 = ( Vtx ‘ ) ) ∧ 𝑒 = ( iEdg ‘ ) ) → 𝒫 𝑣 = 𝒫 ( Vtx ‘ ) )
26 25 difeq1d ( ( ( 𝑔 = 𝑣 = ( Vtx ‘ ) ) ∧ 𝑒 = ( iEdg ‘ ) ) → ( 𝒫 𝑣 ∖ { ∅ } ) = ( 𝒫 ( Vtx ‘ ) ∖ { ∅ } ) )
27 26 rabeqdv ( ( ( 𝑔 = 𝑣 = ( Vtx ‘ ) ) ∧ 𝑒 = ( iEdg ‘ ) ) → { 𝑥 ∈ ( 𝒫 𝑣 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } = { 𝑥 ∈ ( 𝒫 ( Vtx ‘ ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
28 22 23 27 f1eq123d ( ( ( 𝑔 = 𝑣 = ( Vtx ‘ ) ) ∧ 𝑒 = ( iEdg ‘ ) ) → ( 𝑒 : dom 𝑒1-1→ { 𝑥 ∈ ( 𝒫 𝑣 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ↔ ( iEdg ‘ ) : dom ( iEdg ‘ ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
29 19 21 28 sbcied2 ( ( 𝑔 = 𝑣 = ( Vtx ‘ ) ) → ( [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒1-1→ { 𝑥 ∈ ( 𝒫 𝑣 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ↔ ( iEdg ‘ ) : dom ( iEdg ‘ ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
30 17 18 29 sbcied2 ( 𝑔 = → ( [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒1-1→ { 𝑥 ∈ ( 𝒫 𝑣 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ↔ ( iEdg ‘ ) : dom ( iEdg ‘ ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
31 30 cbvabv { 𝑔[ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒1-1→ { 𝑥 ∈ ( 𝒫 𝑣 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } } = { ∣ ( iEdg ‘ ) : dom ( iEdg ‘ ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } }
32 16 31 elab2g ( 𝐺𝑈 → ( 𝐺 ∈ { 𝑔[ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒1-1→ { 𝑥 ∈ ( 𝒫 𝑣 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } } ↔ 𝐸 : dom 𝐸1-1→ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
33 4 32 syl5bb ( 𝐺𝑈 → ( 𝐺 ∈ USPGraph ↔ 𝐸 : dom 𝐸1-1→ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )