Step | Hyp | Ref
| Expression |
1 | | eqid 2738 |
. . . . 5
⊢
(Vtx‘𝐴) =
(Vtx‘𝐴) |
2 | | eqid 2738 |
. . . . 5
⊢
(Vtx‘𝐵) =
(Vtx‘𝐵) |
3 | | eqid 2738 |
. . . . 5
⊢
(iEdg‘𝐴) =
(iEdg‘𝐴) |
4 | | eqid 2738 |
. . . . 5
⊢
(iEdg‘𝐵) =
(iEdg‘𝐵) |
5 | 1, 2, 3, 4 | isomgr 44793 |
. . . 4
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))))) |
6 | 5 | 3adant3 1133 |
. . 3
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))))) |
7 | | eqid 2738 |
. . . . 5
⊢
(Vtx‘𝐶) =
(Vtx‘𝐶) |
8 | | eqid 2738 |
. . . . 5
⊢
(iEdg‘𝐶) =
(iEdg‘𝐶) |
9 | 2, 7, 4, 8 | isomgr 44793 |
. . . 4
⊢ ((𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝐵 IsomGr 𝐶 ↔ ∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))))) |
10 | 9 | 3adant1 1131 |
. . 3
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝐵 IsomGr 𝐶 ↔ ∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))))) |
11 | 6, 10 | anbi12d 634 |
. 2
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → ((𝐴 IsomGr 𝐵 ∧ 𝐵 IsomGr 𝐶) ↔ (∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ ∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))))) |
12 | | vex 3401 |
. . . . . . . . . . 11
⊢ 𝑣 ∈ V |
13 | | vex 3401 |
. . . . . . . . . . 11
⊢ 𝑓 ∈ V |
14 | 12, 13 | coex 7654 |
. . . . . . . . . 10
⊢ (𝑣 ∘ 𝑓) ∈ V |
15 | 14 | a1i 11 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → (𝑣 ∘ 𝑓) ∈ V) |
16 | | simpl 486 |
. . . . . . . . . . 11
⊢ ((𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) |
17 | | simprl 771 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) → 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) |
18 | | f1oco 6634 |
. . . . . . . . . . 11
⊢ ((𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) → (𝑣 ∘ 𝑓):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶)) |
19 | 16, 17, 18 | syl2anr 600 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → (𝑣 ∘ 𝑓):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶)) |
20 | | vex 3401 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑤 ∈ V |
21 | | vex 3401 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑔 ∈ V |
22 | 20, 21 | coex 7654 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 ∘ 𝑔) ∈ V |
23 | 22 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐴 ∈
UHGraph ∧ 𝐵 ∈
UHGraph ∧ 𝐶 ∈
𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → (𝑤 ∘ 𝑔) ∈ V) |
24 | | simpl 486 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → 𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶)) |
25 | | simprl 771 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) → 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵)) |
26 | | f1oco 6634 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵)) →
(𝑤 ∘ 𝑔):dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶)) |
27 | 24, 25, 26 | syl2anr 600 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐴 ∈
UHGraph ∧ 𝐵 ∈
UHGraph ∧ 𝐶 ∈
𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → (𝑤 ∘ 𝑔):dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶)) |
28 | | isomgrtrlem 44808 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐴 ∈
UHGraph ∧ 𝐵 ∈
UHGraph ∧ 𝐶 ∈
𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗))) |
29 | 27, 28 | jca 515 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐴 ∈
UHGraph ∧ 𝐵 ∈
UHGraph ∧ 𝐶 ∈
𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → ((𝑤 ∘ 𝑔):dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗)))) |
30 | | f1oeq1 6600 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℎ = (𝑤 ∘ 𝑔) → (ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ↔
(𝑤 ∘ 𝑔):dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶))) |
31 | | fveq1 6667 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (ℎ = (𝑤 ∘ 𝑔) → (ℎ‘𝑗) = ((𝑤 ∘ 𝑔)‘𝑗)) |
32 | 31 | fveq2d 6672 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (ℎ = (𝑤 ∘ 𝑔) → ((iEdg‘𝐶)‘(ℎ‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗))) |
33 | 32 | eqeq2d 2749 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ℎ = (𝑤 ∘ 𝑔) → (((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)) ↔ ((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗)))) |
34 | 33 | ralbidv 3109 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℎ = (𝑤 ∘ 𝑔) → (∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)) ↔ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗)))) |
35 | 30, 34 | anbi12d 634 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℎ = (𝑤 ∘ 𝑔) → ((ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))) ↔ ((𝑤 ∘ 𝑔):dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗))))) |
36 | 23, 29, 35 | spcedv 3500 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐴 ∈
UHGraph ∧ 𝐵 ∈
UHGraph ∧ 𝐶 ∈
𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))) |
37 | 36 | ex 416 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) → ((𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))) |
38 | 37 | exlimdv 1939 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) → (∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))) |
39 | 38 | ex 416 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) → ((𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))) |
40 | 39 | exlimdv 1939 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) → (∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))) |
41 | 40 | 3exp 1120 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) → (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) → (∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))))) |
42 | 41 | com34 91 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) → (∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) → (∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))))) |
43 | 42 | imp32 422 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) → (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) → (∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))) |
44 | 43 | imp32 422 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))) |
45 | 19, 44 | jca 515 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → ((𝑣 ∘ 𝑓):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))) |
46 | | f1oeq1 6600 |
. . . . . . . . . 10
⊢ (𝑒 = (𝑣 ∘ 𝑓) → (𝑒:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ↔ (𝑣 ∘ 𝑓):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶))) |
47 | | imaeq1 5892 |
. . . . . . . . . . . . . 14
⊢ (𝑒 = (𝑣 ∘ 𝑓) → (𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗))) |
48 | 47 | eqeq1d 2740 |
. . . . . . . . . . . . 13
⊢ (𝑒 = (𝑣 ∘ 𝑓) → ((𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)) ↔ ((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))) |
49 | 48 | ralbidv 3109 |
. . . . . . . . . . . 12
⊢ (𝑒 = (𝑣 ∘ 𝑓) → (∀𝑗 ∈ dom (iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)) ↔ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))) |
50 | 49 | anbi2d 632 |
. . . . . . . . . . 11
⊢ (𝑒 = (𝑣 ∘ 𝑓) → ((ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))) ↔ (ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))) |
51 | 50 | exbidv 1927 |
. . . . . . . . . 10
⊢ (𝑒 = (𝑣 ∘ 𝑓) → (∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))) ↔ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))) |
52 | 46, 51 | anbi12d 634 |
. . . . . . . . 9
⊢ (𝑒 = (𝑣 ∘ 𝑓) → ((𝑒:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))) ↔ ((𝑣 ∘ 𝑓):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))) |
53 | 15, 45, 52 | spcedv 3500 |
. . . . . . . 8
⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → ∃𝑒(𝑒:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))) |
54 | 1, 7, 3, 8 | isomgr 44793 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝐴 IsomGr 𝐶 ↔ ∃𝑒(𝑒:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))) |
55 | 54 | 3adant2 1132 |
. . . . . . . . 9
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝐴 IsomGr 𝐶 ↔ ∃𝑒(𝑒:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))) |
56 | 55 | ad2antrr 726 |
. . . . . . . 8
⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → (𝐴 IsomGr 𝐶 ↔ ∃𝑒(𝑒:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑗 ∈ dom
(iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))) |
57 | 53, 56 | mpbird 260 |
. . . . . . 7
⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → 𝐴 IsomGr 𝐶) |
58 | 57 | ex 416 |
. . . . . 6
⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) → ((𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → 𝐴 IsomGr 𝐶)) |
59 | 58 | exlimdv 1939 |
. . . . 5
⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) → (∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → 𝐴 IsomGr 𝐶)) |
60 | 59 | ex 416 |
. . . 4
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → ((𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) → (∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → 𝐴 IsomGr 𝐶))) |
61 | 60 | exlimdv 1939 |
. . 3
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) → (∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → 𝐴 IsomGr 𝐶))) |
62 | 61 | impd 414 |
. 2
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → ((∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom
(iEdg‘𝐵) ∧
∀𝑖 ∈ dom
(iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ ∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom
(iEdg‘𝐶) ∧
∀𝑘 ∈ dom
(iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → 𝐴 IsomGr 𝐶)) |
63 | 11, 62 | sylbid 243 |
1
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → ((𝐴 IsomGr 𝐵 ∧ 𝐵 IsomGr 𝐶) → 𝐴 IsomGr 𝐶)) |