grimuhgr - Mathbox for Alexander van der Vekens

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Theorem grimuhgr 48508

Description: If there is a graph isomorphism between a hypergraph and a class with an edge function, the class is also a hypergraph. (Contributed by AV, 2-May-2025.)

**Assertion**
Ref	Expression
grimuhgr	⊢ ((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇) ∧ Fun (iEdg‘𝑇)) → 𝑇 ∈ UHGraph)

Proof of Theorem grimuhgr Dummy variables 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . . . 7 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2765	. . . . . . 7 ⊢ (Vtx‘𝑇) = (Vtx‘𝑇)
3		eqid 2765	. . . . . . 7 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2765	. . . . . . 7 ⊢ (iEdg‘𝑇) = (iEdg‘𝑇)
5	1, 2, 3, 4	grimprop 48504	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GraphIso 𝑇) → (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))))
6		fdmrn 6727	. . . . . . . . . . . . . 14 ⊢ (Fun (iEdg‘𝑇) ↔ (iEdg‘𝑇):dom (iEdg‘𝑇)⟶ran (iEdg‘𝑇))
7	6	biimpi 219	. . . . . . . . . . . . 13 ⊢ (Fun (iEdg‘𝑇) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶ran (iEdg‘𝑇))
8	7	3ad2ant3 1151	. . . . . . . . . . . 12 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) ∧ Fun (iEdg‘𝑇)) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶ran (iEdg‘𝑇))
9	8	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) ∧ Fun (iEdg‘𝑇)) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶ran (iEdg‘𝑇))
10		funfn 6555	. . . . . . . . . . . . . 14 ⊢ (Fun (iEdg‘𝑇) ↔ (iEdg‘𝑇) Fn dom (iEdg‘𝑇))
11	10	biimpi 219	. . . . . . . . . . . . 13 ⊢ (Fun (iEdg‘𝑇) → (iEdg‘𝑇) Fn dom (iEdg‘𝑇))
12	11	3ad2ant3 1151	. . . . . . . . . . . 12 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) ∧ Fun (iEdg‘𝑇)) → (iEdg‘𝑇) Fn dom (iEdg‘𝑇))
13		f1ofo 6818	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) → 𝑗:dom (iEdg‘𝑆)–onto→dom (iEdg‘𝑇))
14	13	3ad2ant2 1150	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → 𝑗:dom (iEdg‘𝑆)–onto→dom (iEdg‘𝑇))
15	14	3ad2ant1 1149	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) → 𝑗:dom (iEdg‘𝑆)–onto→dom (iEdg‘𝑇))
16		foelcdmi 6932	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑗:dom (iEdg‘𝑆)–onto→dom (iEdg‘𝑇) ∧ 𝑥 ∈ dom (iEdg‘𝑇)) → ∃𝑦 ∈ dom (iEdg‘𝑆)(𝑗‘𝑦) = 𝑥)
17	15, 16	sylan 591	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) ∧ 𝑥 ∈ dom (iEdg‘𝑇)) → ∃𝑦 ∈ dom (iEdg‘𝑆)(𝑗‘𝑦) = 𝑥)
18	17	ex 417	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) → (𝑥 ∈ dom (iEdg‘𝑇) → ∃𝑦 ∈ dom (iEdg‘𝑆)(𝑗‘𝑦) = 𝑥))
19		2fveq3 6876	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 = 𝑦 → ((iEdg‘𝑇)‘(𝑗‘𝑖)) = ((iEdg‘𝑇)‘(𝑗‘𝑦)))
20		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 = 𝑦 → ((iEdg‘𝑆)‘𝑖) = ((iEdg‘𝑆)‘𝑦))
21	20	imaeq2d 6052	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 = 𝑦 → (𝐹 “ ((iEdg‘𝑆)‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦)))
22	19, 21	eqeq12d 2781	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 = 𝑦 → (((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) ↔ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))))
23	22	rspcv 3580	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ dom (iEdg‘𝑆) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))))
24	23	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))))
25		f1ofun 6812	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → Fun 𝐹)
26	25	3ad2ant1 1149	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → Fun 𝐹)
27	26	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) → Fun 𝐹)
28		fvex 6884	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((iEdg‘𝑆)‘𝑦) ∈ V
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑦) ∈ V)
30		funimaexg 6612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((Fun 𝐹 ∧ ((iEdg‘𝑆)‘𝑦) ∈ V) → (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ∈ V)
31	27, 29, 30	syl2an2r 697	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ∈ V)
32		f1of 6810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → 𝐹:(Vtx‘𝑆)⟶(Vtx‘𝑇))
33	32	fimassd 6717	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ⊆ (Vtx‘𝑇))
34	33	3ad2ant1 1149	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ⊆ (Vtx‘𝑇))
35	34	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) → (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ⊆ (Vtx‘𝑇))
36	35	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ⊆ (Vtx‘𝑇))
37	31, 36	elpwd 4564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ∈ 𝒫 (Vtx‘𝑇))
38		f1odm 6814	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → dom 𝐹 = (Vtx‘𝑆))
39	38	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → dom 𝐹 = (Vtx‘𝑆))
40	39	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → dom 𝐹 = (Vtx‘𝑆))
41	40	ineq1d 4174	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (dom 𝐹 ∩ ((iEdg‘𝑆)‘𝑦)) = ((Vtx‘𝑆) ∩ ((iEdg‘𝑆)‘𝑦)))
42		ffvelcdm 7066	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑦) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}))
43	42	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → (𝑦 ∈ dom (iEdg‘𝑆) → ((iEdg‘𝑆)‘𝑦) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅})))
44	43	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → (𝑦 ∈ dom (iEdg‘𝑆) → ((iEdg‘𝑆)‘𝑦) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅})))
45		eldifsn 4749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((iEdg‘𝑆)‘𝑦) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ↔ (((iEdg‘𝑆)‘𝑦) ∈ 𝒫 (Vtx‘𝑆) ∧ ((iEdg‘𝑆)‘𝑦) ≠ ∅))
46	28	elpw 4562	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((iEdg‘𝑆)‘𝑦) ∈ 𝒫 (Vtx‘𝑆) ↔ ((iEdg‘𝑆)‘𝑦) ⊆ (Vtx‘𝑆))
47	45, 46	bianbi 638	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((iEdg‘𝑆)‘𝑦) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ↔ (((iEdg‘𝑆)‘𝑦) ⊆ (Vtx‘𝑆) ∧ ((iEdg‘𝑆)‘𝑦) ≠ ∅))
48		sseqin2 4178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (((iEdg‘𝑆)‘𝑦) ⊆ (Vtx‘𝑆) ↔ ((Vtx‘𝑆) ∩ ((iEdg‘𝑆)‘𝑦)) = ((iEdg‘𝑆)‘𝑦))
49	48	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((iEdg‘𝑆)‘𝑦) ⊆ (Vtx‘𝑆) → ((Vtx‘𝑆) ∩ ((iEdg‘𝑆)‘𝑦)) = ((iEdg‘𝑆)‘𝑦))
50	49	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((iEdg‘𝑆)‘𝑦) ⊆ (Vtx‘𝑆) ∧ ((iEdg‘𝑆)‘𝑦) ≠ ∅) → ((Vtx‘𝑆) ∩ ((iEdg‘𝑆)‘𝑦)) = ((iEdg‘𝑆)‘𝑦))
51		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((iEdg‘𝑆)‘𝑦) ⊆ (Vtx‘𝑆) ∧ ((iEdg‘𝑆)‘𝑦) ≠ ∅) → ((iEdg‘𝑆)‘𝑦) ≠ ∅)
52	50, 51	eqnetrd 3027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((iEdg‘𝑆)‘𝑦) ⊆ (Vtx‘𝑆) ∧ ((iEdg‘𝑆)‘𝑦) ≠ ∅) → ((Vtx‘𝑆) ∩ ((iEdg‘𝑆)‘𝑦)) ≠ ∅)
53	52	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → ((((iEdg‘𝑆)‘𝑦) ⊆ (Vtx‘𝑆) ∧ ((iEdg‘𝑆)‘𝑦) ≠ ∅) → ((Vtx‘𝑆) ∩ ((iEdg‘𝑆)‘𝑦)) ≠ ∅))
54	47, 53	biimtrid 245	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → (((iEdg‘𝑆)‘𝑦) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) → ((Vtx‘𝑆) ∩ ((iEdg‘𝑆)‘𝑦)) ≠ ∅))
55	44, 54	syld 48	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → (𝑦 ∈ dom (iEdg‘𝑆) → ((Vtx‘𝑆) ∩ ((iEdg‘𝑆)‘𝑦)) ≠ ∅))
56	55	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → ((Vtx‘𝑆) ∩ ((iEdg‘𝑆)‘𝑦)) ≠ ∅)
57	41, 56	eqnetrd 3027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (dom 𝐹 ∩ ((iEdg‘𝑆)‘𝑦)) ≠ ∅)
58	57	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → (𝑦 ∈ dom (iEdg‘𝑆) → (dom 𝐹 ∩ ((iEdg‘𝑆)‘𝑦)) ≠ ∅))
59	58	3adant2 1147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → (𝑦 ∈ dom (iEdg‘𝑆) → (dom 𝐹 ∩ ((iEdg‘𝑆)‘𝑦)) ≠ ∅))
60	59	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) → (𝑦 ∈ dom (iEdg‘𝑆) → (dom 𝐹 ∩ ((iEdg‘𝑆)‘𝑦)) ≠ ∅))
61	60	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (dom 𝐹 ∩ ((iEdg‘𝑆)‘𝑦)) ≠ ∅)
62	61	imadisjlnd 6073	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ≠ ∅)
63		eldifsn 4749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐹 “ ((iEdg‘𝑆)‘𝑦)) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}) ↔ ((𝐹 “ ((iEdg‘𝑆)‘𝑦)) ∈ 𝒫 (Vtx‘𝑇) ∧ (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ≠ ∅))
64	37, 62, 63	sylanbrc 594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}))
65	64	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))) → (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}))
66		eleq1 2853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦)) → (((iEdg‘𝑇)‘(𝑗‘𝑦)) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}) ↔ (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅})))
67	66	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))) → (((iEdg‘𝑇)‘(𝑗‘𝑦)) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}) ↔ (𝐹 “ ((iEdg‘𝑆)‘𝑦)) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅})))
68	65, 67	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))) → ((iEdg‘𝑇)‘(𝑗‘𝑦)) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}))
69		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑇)‘(𝑗‘𝑦)) = ((iEdg‘𝑇)‘𝑥))
70	69	eleq1d 2850	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑗‘𝑦) = 𝑥 → (((iEdg‘𝑇)‘(𝑗‘𝑦)) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}) ↔ ((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅})))
71	68, 70	syl5ibcom 248	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))) → ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅})))
72	71	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦)) → ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}))))
73	24, 72	syld 48	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}))))
74	73	ex 417	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) → (𝑦 ∈ dom (iEdg‘𝑆) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅})))))
75	74	com23 87	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇)) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → (𝑦 ∈ dom (iEdg‘𝑆) → ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅})))))
76	75	ex 417	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → (Fun (iEdg‘𝑇) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → (𝑦 ∈ dom (iEdg‘𝑆) → ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}))))))
77	76	3imp 1126	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) → (𝑦 ∈ dom (iEdg‘𝑆) → ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}))))
78	77	rexlimdv 3164	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) → (∃𝑦 ∈ dom (iEdg‘𝑆)(𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅})))
79	18, 78	syld 48	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) → (𝑥 ∈ dom (iEdg‘𝑇) → ((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅})))
80	79	ralrimiv 3156	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) ∧ Fun (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) → ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}))
81	80	3exp 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → (Fun (iEdg‘𝑇) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}))))
82	81	3exp 1135	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) → ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → (Fun (iEdg‘𝑇) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}))))))
83	82	com35 99	. . . . . . . . . . . . . . 15 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → (Fun (iEdg‘𝑇) → ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}))))))
84	83	impd 415	. . . . . . . . . . . . . 14 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → ((𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) → (Fun (iEdg‘𝑇) → ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅})))))
85	84	3imp 1126	. . . . . . . . . . . . 13 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) ∧ Fun (iEdg‘𝑇)) → ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅})))
86	85	imp 411	. . . . . . . . . . . 12 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) ∧ Fun (iEdg‘𝑇)) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅}))
87		fnfvrnss 7106	. . . . . . . . . . . 12 ⊢ (((iEdg‘𝑇) Fn dom (iEdg‘𝑇) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑇)‘𝑥) ∈ (𝒫 (Vtx‘𝑇) ∖ {∅})) → ran (iEdg‘𝑇) ⊆ (𝒫 (Vtx‘𝑇) ∖ {∅}))
88	12, 86, 87	syl2an2r 697	. . . . . . . . . . 11 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) ∧ Fun (iEdg‘𝑇)) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → ran (iEdg‘𝑇) ⊆ (𝒫 (Vtx‘𝑇) ∖ {∅}))
89	9, 88	fssd 6713	. . . . . . . . . 10 ⊢ (((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) ∧ Fun (iEdg‘𝑇)) ∧ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶(𝒫 (Vtx‘𝑇) ∖ {∅}))
90	89	ex 417	. . . . . . . . 9 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) ∧ Fun (iEdg‘𝑇)) → ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶(𝒫 (Vtx‘𝑇) ∖ {∅})))
91	90	3exp 1135	. . . . . . . 8 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → ((𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) → (Fun (iEdg‘𝑇) → ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶(𝒫 (Vtx‘𝑇) ∖ {∅})))))
92	91	exlimdv 1956	. . . . . . 7 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → (∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) → (Fun (iEdg‘𝑇) → ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶(𝒫 (Vtx‘𝑇) ∖ {∅})))))
93	92	imp 411	. . . . . 6 ⊢ ((𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) → (Fun (iEdg‘𝑇) → ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶(𝒫 (Vtx‘𝑇) ∖ {∅}))))
94	5, 93	syl 18	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GraphIso 𝑇) → (Fun (iEdg‘𝑇) → ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶(𝒫 (Vtx‘𝑇) ∖ {∅}))))
95	94	impcom 412	. . . 4 ⊢ ((Fun (iEdg‘𝑇) ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) → ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶(𝒫 (Vtx‘𝑇) ∖ {∅})))
96		grimdmrel 48501	. . . . . . 7 ⊢ Rel dom GraphIso
97	96	ovrcl 7441	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GraphIso 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
98	1, 3	isuhgr 29315	. . . . . . . 8 ⊢ (𝑆 ∈ V → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})))
99	98	adantr 485	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})))
100	2, 4	isuhgr 29315	. . . . . . . 8 ⊢ (𝑇 ∈ V → (𝑇 ∈ UHGraph ↔ (iEdg‘𝑇):dom (iEdg‘𝑇)⟶(𝒫 (Vtx‘𝑇) ∖ {∅})))
101	100	adantl 486	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑇 ∈ UHGraph ↔ (iEdg‘𝑇):dom (iEdg‘𝑇)⟶(𝒫 (Vtx‘𝑇) ∖ {∅})))
102	99, 101	imbi12d 347	. . . . . 6 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → ((𝑆 ∈ UHGraph → 𝑇 ∈ UHGraph) ↔ ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶(𝒫 (Vtx‘𝑇) ∖ {∅}))))
103	97, 102	syl 18	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GraphIso 𝑇) → ((𝑆 ∈ UHGraph → 𝑇 ∈ UHGraph) ↔ ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶(𝒫 (Vtx‘𝑇) ∖ {∅}))))
104	103	adantl 486	. . . 4 ⊢ ((Fun (iEdg‘𝑇) ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) → ((𝑆 ∈ UHGraph → 𝑇 ∈ UHGraph) ↔ ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) → (iEdg‘𝑇):dom (iEdg‘𝑇)⟶(𝒫 (Vtx‘𝑇) ∖ {∅}))))
105	95, 104	mpbird 260	. . 3 ⊢ ((Fun (iEdg‘𝑇) ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) → (𝑆 ∈ UHGraph → 𝑇 ∈ UHGraph))
106	105	ex 417	. 2 ⊢ (Fun (iEdg‘𝑇) → (𝐹 ∈ (𝑆 GraphIso 𝑇) → (𝑆 ∈ UHGraph → 𝑇 ∈ UHGraph)))
107	106	3imp31 1127	1 ⊢ ((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇) ∧ Fun (iEdg‘𝑇)) → 𝑇 ∈ UHGraph)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∃wex 1802 ∈ wcel 2145 ≠ wne 2960 ∀wral 3079 ∃wrex 3089 Vcvv 3457 ∖ cdif 3904 ∩ cin 3906 ⊆ wss 3907 ∅c0 4288 𝒫 cpw 4558 {csn 4585 dom cdm 5651 ran crn 5652 “ cima 5654 Fun wfun 6519 Fn wfn 6520 ⟶wf 6521 –onto→wfo 6523 –1-1-onto→wf1o 6524 ‘cfv 6525 (class class class)co 7400 Vtxcvtx 29251 iEdgciedg 29252 UHGraphcuhgr 29311 GraphIso cgrim 48496

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-sbc 3748 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5105 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-ov 7403 df-oprab 7404 df-mpo 7405 df-map 8814 df-uhgr 29313 df-grim 48499

This theorem is referenced by: (None)

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