# Metamath Proof Explorer

## Theorem isomuspgrlem2b

Description: Lemma 2 for isomuspgrlem2 . (Contributed by AV, 29-Nov-2022)

Ref Expression
Hypotheses isomushgr.v 𝑉 = ( Vtx ‘ 𝐴 )
isomushgr.w 𝑊 = ( Vtx ‘ 𝐵 )
isomushgr.e 𝐸 = ( Edg ‘ 𝐴 )
isomushgr.k 𝐾 = ( Edg ‘ 𝐵 )
isomuspgrlem2.g 𝐺 = ( 𝑥𝐸 ↦ ( 𝐹𝑥 ) )
isomuspgrlem2.a ( 𝜑𝐴 ∈ USPGraph )
isomuspgrlem2.f ( 𝜑𝐹 : 𝑉1-1-onto𝑊 )
isomuspgrlem2.i ( 𝜑 → ∀ 𝑎𝑉𝑏𝑉 ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) )
Assertion isomuspgrlem2b ( 𝜑𝐺 : 𝐸𝐾 )

### Proof

Step Hyp Ref Expression
1 isomushgr.v 𝑉 = ( Vtx ‘ 𝐴 )
2 isomushgr.w 𝑊 = ( Vtx ‘ 𝐵 )
3 isomushgr.e 𝐸 = ( Edg ‘ 𝐴 )
4 isomushgr.k 𝐾 = ( Edg ‘ 𝐵 )
5 isomuspgrlem2.g 𝐺 = ( 𝑥𝐸 ↦ ( 𝐹𝑥 ) )
6 isomuspgrlem2.a ( 𝜑𝐴 ∈ USPGraph )
7 isomuspgrlem2.f ( 𝜑𝐹 : 𝑉1-1-onto𝑊 )
8 isomuspgrlem2.i ( 𝜑 → ∀ 𝑎𝑉𝑏𝑉 ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) )
9 uspgrupgr ( 𝐴 ∈ USPGraph → 𝐴 ∈ UPGraph )
10 6 9 syl ( 𝜑𝐴 ∈ UPGraph )
11 1 3 upgredg ( ( 𝐴 ∈ UPGraph ∧ 𝑥𝐸 ) → ∃ 𝑐𝑉𝑑𝑉 𝑥 = { 𝑐 , 𝑑 } )
12 10 11 sylan ( ( 𝜑𝑥𝐸 ) → ∃ 𝑐𝑉𝑑𝑉 𝑥 = { 𝑐 , 𝑑 } )
13 preq12 ( ( 𝑎 = 𝑐𝑏 = 𝑑 ) → { 𝑎 , 𝑏 } = { 𝑐 , 𝑑 } )
14 13 eleq1d ( ( 𝑎 = 𝑐𝑏 = 𝑑 ) → ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { 𝑐 , 𝑑 } ∈ 𝐸 ) )
15 fveq2 ( 𝑎 = 𝑐 → ( 𝐹𝑎 ) = ( 𝐹𝑐 ) )
16 15 adantr ( ( 𝑎 = 𝑐𝑏 = 𝑑 ) → ( 𝐹𝑎 ) = ( 𝐹𝑐 ) )
17 fveq2 ( 𝑏 = 𝑑 → ( 𝐹𝑏 ) = ( 𝐹𝑑 ) )
18 17 adantl ( ( 𝑎 = 𝑐𝑏 = 𝑑 ) → ( 𝐹𝑏 ) = ( 𝐹𝑑 ) )
19 16 18 preq12d ( ( 𝑎 = 𝑐𝑏 = 𝑑 ) → { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } = { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } )
20 19 eleq1d ( ( 𝑎 = 𝑐𝑏 = 𝑑 ) → ( { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ↔ { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ) )
21 14 20 bibi12d ( ( 𝑎 = 𝑐𝑏 = 𝑑 ) → ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) ↔ ( { 𝑐 , 𝑑 } ∈ 𝐸 ↔ { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ) ) )
22 21 rspc2gv ( ( 𝑐𝑉𝑑𝑉 ) → ( ∀ 𝑎𝑉𝑏𝑉 ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) → ( { 𝑐 , 𝑑 } ∈ 𝐸 ↔ { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ) ) )
23 8 22 syl5com ( 𝜑 → ( ( 𝑐𝑉𝑑𝑉 ) → ( { 𝑐 , 𝑑 } ∈ 𝐸 ↔ { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ) ) )
24 23 adantr ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) → ( ( 𝑐𝑉𝑑𝑉 ) → ( { 𝑐 , 𝑑 } ∈ 𝐸 ↔ { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ) ) )
25 24 imp ( ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → ( { 𝑐 , 𝑑 } ∈ 𝐸 ↔ { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ) )
26 bicom ( ( { 𝑐 , 𝑑 } ∈ 𝐸 ↔ { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ) ↔ ( { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ↔ { 𝑐 , 𝑑 } ∈ 𝐸 ) )
27 bianir ( ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ ( { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ↔ { 𝑐 , 𝑑 } ∈ 𝐸 ) ) → { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 )
28 27 ex ( { 𝑐 , 𝑑 } ∈ 𝐸 → ( ( { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ↔ { 𝑐 , 𝑑 } ∈ 𝐸 ) → { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ) )
29 26 28 syl5bi ( { 𝑐 , 𝑑 } ∈ 𝐸 → ( ( { 𝑐 , 𝑑 } ∈ 𝐸 ↔ { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ) → { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ) )
30 f1ofn ( 𝐹 : 𝑉1-1-onto𝑊𝐹 Fn 𝑉 )
31 7 30 syl ( 𝜑𝐹 Fn 𝑉 )
32 31 adantr ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) → 𝐹 Fn 𝑉 )
33 32 adantr ( ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → 𝐹 Fn 𝑉 )
34 simprl ( ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → 𝑐𝑉 )
35 simprr ( ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → 𝑑𝑉 )
36 33 34 35 3jca ( ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → ( 𝐹 Fn 𝑉𝑐𝑉𝑑𝑉 ) )
37 36 adantl ( ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) ) → ( 𝐹 Fn 𝑉𝑐𝑉𝑑𝑉 ) )
38 fnimapr ( ( 𝐹 Fn 𝑉𝑐𝑉𝑑𝑉 ) → ( 𝐹 “ { 𝑐 , 𝑑 } ) = { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } )
39 37 38 syl ( ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) ) → ( 𝐹 “ { 𝑐 , 𝑑 } ) = { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } )
40 39 eqcomd ( ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) ) → { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } = ( 𝐹 “ { 𝑐 , 𝑑 } ) )
41 40 eleq1d ( ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) ) → ( { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ↔ ( 𝐹 “ { 𝑐 , 𝑑 } ) ∈ 𝐾 ) )
42 41 biimpd ( ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) ) → ( { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 → ( 𝐹 “ { 𝑐 , 𝑑 } ) ∈ 𝐾 ) )
43 42 ex ( { 𝑐 , 𝑑 } ∈ 𝐸 → ( ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → ( { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 → ( 𝐹 “ { 𝑐 , 𝑑 } ) ∈ 𝐾 ) ) )
44 43 com23 ( { 𝑐 , 𝑑 } ∈ 𝐸 → ( { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 → ( ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → ( 𝐹 “ { 𝑐 , 𝑑 } ) ∈ 𝐾 ) ) )
45 29 44 syld ( { 𝑐 , 𝑑 } ∈ 𝐸 → ( ( { 𝑐 , 𝑑 } ∈ 𝐸 ↔ { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ) → ( ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → ( 𝐹 “ { 𝑐 , 𝑑 } ) ∈ 𝐾 ) ) )
46 45 com13 ( ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → ( ( { 𝑐 , 𝑑 } ∈ 𝐸 ↔ { ( 𝐹𝑐 ) , ( 𝐹𝑑 ) } ∈ 𝐾 ) → ( { 𝑐 , 𝑑 } ∈ 𝐸 → ( 𝐹 “ { 𝑐 , 𝑑 } ) ∈ 𝐾 ) ) )
47 25 46 mpd ( ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → ( { 𝑐 , 𝑑 } ∈ 𝐸 → ( 𝐹 “ { 𝑐 , 𝑑 } ) ∈ 𝐾 ) )
48 eleq1 ( 𝑥 = { 𝑐 , 𝑑 } → ( 𝑥𝐸 ↔ { 𝑐 , 𝑑 } ∈ 𝐸 ) )
49 imaeq2 ( 𝑥 = { 𝑐 , 𝑑 } → ( 𝐹𝑥 ) = ( 𝐹 “ { 𝑐 , 𝑑 } ) )
50 49 eleq1d ( 𝑥 = { 𝑐 , 𝑑 } → ( ( 𝐹𝑥 ) ∈ 𝐾 ↔ ( 𝐹 “ { 𝑐 , 𝑑 } ) ∈ 𝐾 ) )
51 48 50 imbi12d ( 𝑥 = { 𝑐 , 𝑑 } → ( ( 𝑥𝐸 → ( 𝐹𝑥 ) ∈ 𝐾 ) ↔ ( { 𝑐 , 𝑑 } ∈ 𝐸 → ( 𝐹 “ { 𝑐 , 𝑑 } ) ∈ 𝐾 ) ) )
52 51 adantl ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) → ( ( 𝑥𝐸 → ( 𝐹𝑥 ) ∈ 𝐾 ) ↔ ( { 𝑐 , 𝑑 } ∈ 𝐸 → ( 𝐹 “ { 𝑐 , 𝑑 } ) ∈ 𝐾 ) ) )
53 52 adantr ( ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → ( ( 𝑥𝐸 → ( 𝐹𝑥 ) ∈ 𝐾 ) ↔ ( { 𝑐 , 𝑑 } ∈ 𝐸 → ( 𝐹 “ { 𝑐 , 𝑑 } ) ∈ 𝐾 ) ) )
54 47 53 mpbird ( ( ( 𝜑𝑥 = { 𝑐 , 𝑑 } ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → ( 𝑥𝐸 → ( 𝐹𝑥 ) ∈ 𝐾 ) )
55 54 exp31 ( 𝜑 → ( 𝑥 = { 𝑐 , 𝑑 } → ( ( 𝑐𝑉𝑑𝑉 ) → ( 𝑥𝐸 → ( 𝐹𝑥 ) ∈ 𝐾 ) ) ) )
56 55 com24 ( 𝜑 → ( 𝑥𝐸 → ( ( 𝑐𝑉𝑑𝑉 ) → ( 𝑥 = { 𝑐 , 𝑑 } → ( 𝐹𝑥 ) ∈ 𝐾 ) ) ) )
57 56 imp31 ( ( ( 𝜑𝑥𝐸 ) ∧ ( 𝑐𝑉𝑑𝑉 ) ) → ( 𝑥 = { 𝑐 , 𝑑 } → ( 𝐹𝑥 ) ∈ 𝐾 ) )
58 57 rexlimdvva ( ( 𝜑𝑥𝐸 ) → ( ∃ 𝑐𝑉𝑑𝑉 𝑥 = { 𝑐 , 𝑑 } → ( 𝐹𝑥 ) ∈ 𝐾 ) )
59 12 58 mpd ( ( 𝜑𝑥𝐸 ) → ( 𝐹𝑥 ) ∈ 𝐾 )
60 59 ralrimiva ( 𝜑 → ∀ 𝑥𝐸 ( 𝐹𝑥 ) ∈ 𝐾 )
61 5 fmpt ( ∀ 𝑥𝐸 ( 𝐹𝑥 ) ∈ 𝐾𝐺 : 𝐸𝐾 )
62 60 61 sylib ( 𝜑𝐺 : 𝐸𝐾 )