# Metamath Proof Explorer

## Theorem isomuspgrlem2d

Description: Lemma 4 for isomuspgrlem2 . (Contributed by AV, 1-Dec-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 ) )
isomuspgrlem2.x ( 𝜑𝐹𝑋 )
isomuspgrlem2.b ( 𝜑𝐵 ∈ USPGraph )
Assertion isomuspgrlem2d ( 𝜑𝐺 : 𝐸onto𝐾 )

### 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 isomuspgrlem2.x ( 𝜑𝐹𝑋 )
10 isomuspgrlem2.b ( 𝜑𝐵 ∈ USPGraph )
11 1 2 3 4 5 6 7 8 isomuspgrlem2b ( 𝜑𝐺 : 𝐸𝐾 )
12 uspgrupgr ( 𝐵 ∈ USPGraph → 𝐵 ∈ UPGraph )
13 10 12 syl ( 𝜑𝐵 ∈ UPGraph )
14 2 4 upgredg ( ( 𝐵 ∈ UPGraph ∧ 𝑦𝐾 ) → ∃ 𝑐𝑊𝑑𝑊 𝑦 = { 𝑐 , 𝑑 } )
15 13 14 sylan ( ( 𝜑𝑦𝐾 ) → ∃ 𝑐𝑊𝑑𝑊 𝑦 = { 𝑐 , 𝑑 } )
16 eleq1 ( 𝑦 = { 𝑐 , 𝑑 } → ( 𝑦𝐾 ↔ { 𝑐 , 𝑑 } ∈ 𝐾 ) )
17 16 anbi2d ( 𝑦 = { 𝑐 , 𝑑 } → ( ( 𝜑𝑦𝐾 ) ↔ ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ) )
18 f1ofo ( 𝐹 : 𝑉1-1-onto𝑊𝐹 : 𝑉onto𝑊 )
19 7 18 syl ( 𝜑𝐹 : 𝑉onto𝑊 )
20 foelrn ( ( 𝐹 : 𝑉onto𝑊𝑐𝑊 ) → ∃ 𝑚𝑉 𝑐 = ( 𝐹𝑚 ) )
21 19 20 sylan ( ( 𝜑𝑐𝑊 ) → ∃ 𝑚𝑉 𝑐 = ( 𝐹𝑚 ) )
22 21 ex ( 𝜑 → ( 𝑐𝑊 → ∃ 𝑚𝑉 𝑐 = ( 𝐹𝑚 ) ) )
23 foelrn ( ( 𝐹 : 𝑉onto𝑊𝑑𝑊 ) → ∃ 𝑛𝑉 𝑑 = ( 𝐹𝑛 ) )
24 19 23 sylan ( ( 𝜑𝑑𝑊 ) → ∃ 𝑛𝑉 𝑑 = ( 𝐹𝑛 ) )
25 24 ex ( 𝜑 → ( 𝑑𝑊 → ∃ 𝑛𝑉 𝑑 = ( 𝐹𝑛 ) ) )
26 22 25 anim12d ( 𝜑 → ( ( 𝑐𝑊𝑑𝑊 ) → ( ∃ 𝑚𝑉 𝑐 = ( 𝐹𝑚 ) ∧ ∃ 𝑛𝑉 𝑑 = ( 𝐹𝑛 ) ) ) )
27 26 adantr ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) → ( ( 𝑐𝑊𝑑𝑊 ) → ( ∃ 𝑚𝑉 𝑐 = ( 𝐹𝑚 ) ∧ ∃ 𝑛𝑉 𝑑 = ( 𝐹𝑛 ) ) ) )
28 27 imp ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) → ( ∃ 𝑚𝑉 𝑐 = ( 𝐹𝑚 ) ∧ ∃ 𝑛𝑉 𝑑 = ( 𝐹𝑛 ) ) )
29 preq12 ( ( 𝑐 = ( 𝐹𝑚 ) ∧ 𝑑 = ( 𝐹𝑛 ) ) → { 𝑐 , 𝑑 } = { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } )
30 29 ancoms ( ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) → { 𝑐 , 𝑑 } = { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } )
31 30 eleq1d ( ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) → ( { 𝑐 , 𝑑 } ∈ 𝐾 ↔ { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } ∈ 𝐾 ) )
32 preq1 ( 𝑎 = 𝑚 → { 𝑎 , 𝑏 } = { 𝑚 , 𝑏 } )
33 32 eleq1d ( 𝑎 = 𝑚 → ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { 𝑚 , 𝑏 } ∈ 𝐸 ) )
34 fveq2 ( 𝑎 = 𝑚 → ( 𝐹𝑎 ) = ( 𝐹𝑚 ) )
35 34 preq1d ( 𝑎 = 𝑚 → { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } = { ( 𝐹𝑚 ) , ( 𝐹𝑏 ) } )
36 35 eleq1d ( 𝑎 = 𝑚 → ( { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ↔ { ( 𝐹𝑚 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) )
37 33 36 bibi12d ( 𝑎 = 𝑚 → ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) ↔ ( { 𝑚 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑚 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) ) )
38 preq2 ( 𝑏 = 𝑛 → { 𝑚 , 𝑏 } = { 𝑚 , 𝑛 } )
39 38 eleq1d ( 𝑏 = 𝑛 → ( { 𝑚 , 𝑏 } ∈ 𝐸 ↔ { 𝑚 , 𝑛 } ∈ 𝐸 ) )
40 fveq2 ( 𝑏 = 𝑛 → ( 𝐹𝑏 ) = ( 𝐹𝑛 ) )
41 40 preq2d ( 𝑏 = 𝑛 → { ( 𝐹𝑚 ) , ( 𝐹𝑏 ) } = { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } )
42 41 eleq1d ( 𝑏 = 𝑛 → ( { ( 𝐹𝑚 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ↔ { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } ∈ 𝐾 ) )
43 39 42 bibi12d ( 𝑏 = 𝑛 → ( ( { 𝑚 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑚 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) ↔ ( { 𝑚 , 𝑛 } ∈ 𝐸 ↔ { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } ∈ 𝐾 ) ) )
44 37 43 rspc2va ( ( ( 𝑚𝑉𝑛𝑉 ) ∧ ∀ 𝑎𝑉𝑏𝑉 ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) ) → ( { 𝑚 , 𝑛 } ∈ 𝐸 ↔ { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } ∈ 𝐾 ) )
45 44 bicomd ( ( ( 𝑚𝑉𝑛𝑉 ) ∧ ∀ 𝑎𝑉𝑏𝑉 ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) ) → ( { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } ∈ 𝐾 ↔ { 𝑚 , 𝑛 } ∈ 𝐸 ) )
46 45 ancoms ( ( ∀ 𝑎𝑉𝑏𝑉 ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) → ( { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } ∈ 𝐾 ↔ { 𝑚 , 𝑛 } ∈ 𝐸 ) )
47 46 biimpd ( ( ∀ 𝑎𝑉𝑏𝑉 ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) → ( { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } ∈ 𝐾 → { 𝑚 , 𝑛 } ∈ 𝐸 ) )
48 47 ex ( ∀ 𝑎𝑉𝑏𝑉 ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) → ( ( 𝑚𝑉𝑛𝑉 ) → ( { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } ∈ 𝐾 → { 𝑚 , 𝑛 } ∈ 𝐸 ) ) )
49 8 48 syl ( 𝜑 → ( ( 𝑚𝑉𝑛𝑉 ) → ( { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } ∈ 𝐾 → { 𝑚 , 𝑛 } ∈ 𝐸 ) ) )
50 49 com13 ( { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } ∈ 𝐾 → ( ( 𝑚𝑉𝑛𝑉 ) → ( 𝜑 → { 𝑚 , 𝑛 } ∈ 𝐸 ) ) )
51 31 50 syl6bi ( ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) → ( { 𝑐 , 𝑑 } ∈ 𝐾 → ( ( 𝑚𝑉𝑛𝑉 ) → ( 𝜑 → { 𝑚 , 𝑛 } ∈ 𝐸 ) ) ) )
52 51 com14 ( 𝜑 → ( { 𝑐 , 𝑑 } ∈ 𝐾 → ( ( 𝑚𝑉𝑛𝑉 ) → ( ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) → { 𝑚 , 𝑛 } ∈ 𝐸 ) ) ) )
53 52 imp ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) → ( ( 𝑚𝑉𝑛𝑉 ) → ( ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) → { 𝑚 , 𝑛 } ∈ 𝐸 ) ) )
54 53 adantr ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) → ( ( 𝑚𝑉𝑛𝑉 ) → ( ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) → { 𝑚 , 𝑛 } ∈ 𝐸 ) ) )
55 54 imp31 ( ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) ∧ ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) ) → { 𝑚 , 𝑛 } ∈ 𝐸 )
56 imaeq2 ( 𝑒 = { 𝑚 , 𝑛 } → ( 𝐹𝑒 ) = ( 𝐹 “ { 𝑚 , 𝑛 } ) )
57 f1ofn ( 𝐹 : 𝑉1-1-onto𝑊𝐹 Fn 𝑉 )
58 7 57 syl ( 𝜑𝐹 Fn 𝑉 )
59 58 ad3antrrr ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) → 𝐹 Fn 𝑉 )
60 simprl ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) → 𝑚𝑉 )
61 simpr ( ( 𝑚𝑉𝑛𝑉 ) → 𝑛𝑉 )
62 61 adantl ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) → 𝑛𝑉 )
63 59 60 62 3jca ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) → ( 𝐹 Fn 𝑉𝑚𝑉𝑛𝑉 ) )
64 63 adantr ( ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) ∧ ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) ) → ( 𝐹 Fn 𝑉𝑚𝑉𝑛𝑉 ) )
65 fnimapr ( ( 𝐹 Fn 𝑉𝑚𝑉𝑛𝑉 ) → ( 𝐹 “ { 𝑚 , 𝑛 } ) = { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } )
66 64 65 syl ( ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) ∧ ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) ) → ( 𝐹 “ { 𝑚 , 𝑛 } ) = { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } )
67 56 66 sylan9eqr ( ( ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) ∧ ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) ) ∧ 𝑒 = { 𝑚 , 𝑛 } ) → ( 𝐹𝑒 ) = { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } )
68 67 eqeq2d ( ( ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) ∧ ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) ) ∧ 𝑒 = { 𝑚 , 𝑛 } ) → ( { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ↔ { 𝑐 , 𝑑 } = { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } ) )
69 30 adantl ( ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) ∧ ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) ) → { 𝑐 , 𝑑 } = { ( 𝐹𝑚 ) , ( 𝐹𝑛 ) } )
70 55 68 69 rspcedvd ( ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) ∧ ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) )
71 70 ex ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ ( 𝑚𝑉𝑛𝑉 ) ) → ( ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ) )
72 71 anassrs ( ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑚𝑉 ) ∧ 𝑛𝑉 ) → ( ( 𝑑 = ( 𝐹𝑛 ) ∧ 𝑐 = ( 𝐹𝑚 ) ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ) )
73 72 expd ( ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑚𝑉 ) ∧ 𝑛𝑉 ) → ( 𝑑 = ( 𝐹𝑛 ) → ( 𝑐 = ( 𝐹𝑚 ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ) ) )
74 73 rexlimdva ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑚𝑉 ) → ( ∃ 𝑛𝑉 𝑑 = ( 𝐹𝑛 ) → ( 𝑐 = ( 𝐹𝑚 ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ) ) )
75 74 com23 ( ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑚𝑉 ) → ( 𝑐 = ( 𝐹𝑚 ) → ( ∃ 𝑛𝑉 𝑑 = ( 𝐹𝑛 ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ) ) )
76 75 rexlimdva ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) → ( ∃ 𝑚𝑉 𝑐 = ( 𝐹𝑚 ) → ( ∃ 𝑛𝑉 𝑑 = ( 𝐹𝑛 ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ) ) )
77 76 impd ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) → ( ( ∃ 𝑚𝑉 𝑐 = ( 𝐹𝑚 ) ∧ ∃ 𝑛𝑉 𝑑 = ( 𝐹𝑛 ) ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ) )
78 28 77 mpd ( ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) )
79 78 ex ( ( 𝜑 ∧ { 𝑐 , 𝑑 } ∈ 𝐾 ) → ( ( 𝑐𝑊𝑑𝑊 ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ) )
80 17 79 syl6bi ( 𝑦 = { 𝑐 , 𝑑 } → ( ( 𝜑𝑦𝐾 ) → ( ( 𝑐𝑊𝑑𝑊 ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ) ) )
81 80 impd ( 𝑦 = { 𝑐 , 𝑑 } → ( ( ( 𝜑𝑦𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ) )
82 81 impcom ( ( ( ( 𝜑𝑦𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑦 = { 𝑐 , 𝑑 } ) → ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) )
83 simpr ( ( ( ( 𝜑𝑦𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑦 = { 𝑐 , 𝑑 } ) → 𝑦 = { 𝑐 , 𝑑 } )
84 83 adantr ( ( ( ( ( 𝜑𝑦𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑦 = { 𝑐 , 𝑑 } ) ∧ 𝑒𝐸 ) → 𝑦 = { 𝑐 , 𝑑 } )
85 9 ad4antr ( ( ( ( ( 𝜑𝑦𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑦 = { 𝑐 , 𝑑 } ) ∧ 𝑒𝐸 ) → 𝐹𝑋 )
86 1 2 3 4 5 isomuspgrlem2a ( 𝐹𝑋 → ∀ 𝑦𝐸 ( 𝐹𝑦 ) = ( 𝐺𝑦 ) )
87 85 86 syl ( ( ( ( ( 𝜑𝑦𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑦 = { 𝑐 , 𝑑 } ) ∧ 𝑒𝐸 ) → ∀ 𝑦𝐸 ( 𝐹𝑦 ) = ( 𝐺𝑦 ) )
88 imaeq2 ( 𝑦 = 𝑒 → ( 𝐹𝑦 ) = ( 𝐹𝑒 ) )
89 fveq2 ( 𝑦 = 𝑒 → ( 𝐺𝑦 ) = ( 𝐺𝑒 ) )
90 88 89 eqeq12d ( 𝑦 = 𝑒 → ( ( 𝐹𝑦 ) = ( 𝐺𝑦 ) ↔ ( 𝐹𝑒 ) = ( 𝐺𝑒 ) ) )
91 90 rspcv ( 𝑒𝐸 → ( ∀ 𝑦𝐸 ( 𝐹𝑦 ) = ( 𝐺𝑦 ) → ( 𝐹𝑒 ) = ( 𝐺𝑒 ) ) )
92 eqcom ( ( 𝐺𝑒 ) = ( 𝐹𝑒 ) ↔ ( 𝐹𝑒 ) = ( 𝐺𝑒 ) )
93 91 92 syl6ibr ( 𝑒𝐸 → ( ∀ 𝑦𝐸 ( 𝐹𝑦 ) = ( 𝐺𝑦 ) → ( 𝐺𝑒 ) = ( 𝐹𝑒 ) ) )
94 93 adantl ( ( ( ( ( 𝜑𝑦𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑦 = { 𝑐 , 𝑑 } ) ∧ 𝑒𝐸 ) → ( ∀ 𝑦𝐸 ( 𝐹𝑦 ) = ( 𝐺𝑦 ) → ( 𝐺𝑒 ) = ( 𝐹𝑒 ) ) )
95 87 94 mpd ( ( ( ( ( 𝜑𝑦𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑦 = { 𝑐 , 𝑑 } ) ∧ 𝑒𝐸 ) → ( 𝐺𝑒 ) = ( 𝐹𝑒 ) )
96 84 95 eqeq12d ( ( ( ( ( 𝜑𝑦𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑦 = { 𝑐 , 𝑑 } ) ∧ 𝑒𝐸 ) → ( 𝑦 = ( 𝐺𝑒 ) ↔ { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ) )
97 96 rexbidva ( ( ( ( 𝜑𝑦𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑦 = { 𝑐 , 𝑑 } ) → ( ∃ 𝑒𝐸 𝑦 = ( 𝐺𝑒 ) ↔ ∃ 𝑒𝐸 { 𝑐 , 𝑑 } = ( 𝐹𝑒 ) ) )
98 82 97 mpbird ( ( ( ( 𝜑𝑦𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) ∧ 𝑦 = { 𝑐 , 𝑑 } ) → ∃ 𝑒𝐸 𝑦 = ( 𝐺𝑒 ) )
99 98 ex ( ( ( 𝜑𝑦𝐾 ) ∧ ( 𝑐𝑊𝑑𝑊 ) ) → ( 𝑦 = { 𝑐 , 𝑑 } → ∃ 𝑒𝐸 𝑦 = ( 𝐺𝑒 ) ) )
100 99 rexlimdvva ( ( 𝜑𝑦𝐾 ) → ( ∃ 𝑐𝑊𝑑𝑊 𝑦 = { 𝑐 , 𝑑 } → ∃ 𝑒𝐸 𝑦 = ( 𝐺𝑒 ) ) )
101 15 100 mpd ( ( 𝜑𝑦𝐾 ) → ∃ 𝑒𝐸 𝑦 = ( 𝐺𝑒 ) )
102 101 ralrimiva ( 𝜑 → ∀ 𝑦𝐾𝑒𝐸 𝑦 = ( 𝐺𝑒 ) )
103 dffo3 ( 𝐺 : 𝐸onto𝐾 ↔ ( 𝐺 : 𝐸𝐾 ∧ ∀ 𝑦𝐾𝑒𝐸 𝑦 = ( 𝐺𝑒 ) ) )
104 11 102 103 sylanbrc ( 𝜑𝐺 : 𝐸onto𝐾 )