Theorem uhgrimedg 48794

Description: An isomorphism between graphs preserves edges, i.e. there is an edge in one graph connecting vertices iff there is an edge in the other graph connecting the corresponding vertices. (Contributed by AV, 25-Oct-2025.)

Hypotheses

Ref	Expression
uhgrimedgi.e	⊢ 𝐸 = (Edg‘𝐺)
uhgrimedgi.d	⊢ 𝐷 = (Edg‘𝐻)

Assertion

Ref	Expression
uhgrimedg	⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐾 ∈ 𝐸 ↔ (𝐹 “ 𝐾) ∈ 𝐷))

Proof of Theorem uhgrimedg

Step	Hyp	Ref	Expression
1		simp1 1154	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph))
2		simp2 1155	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → 𝐹 ∈ (𝐺 GraphIso 𝐻))
3	2	anim1i 627	. . 3 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ 𝐾 ∈ 𝐸) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸))
4		uhgrimedgi.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
5		uhgrimedgi.d	. . . 4 ⊢ 𝐷 = (Edg‘𝐻)
6	4, 5	uhgrimedgi 48793	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ (𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸)) → (𝐹 “ 𝐾) ∈ 𝐷)
7	1, 3, 6	syl2an2r 698	. 2 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ 𝐾 ∈ 𝐸) → (𝐹 “ 𝐾) ∈ 𝐷)
8		eqid 2762	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2762	. . . . . . . . 9 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
10	8, 9	grimf1o 48787	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
11		f1of1 6820	. . . . . . . 8 ⊢ (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → 𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
12	10, 11	syl 18	. . . . . . 7 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
13	12	3ad2ant2 1152	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → 𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
14		simp3 1156	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → 𝐾 ⊆ (Vtx‘𝐺))
15	13, 14	jca 521	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)))
16	15	adantr 486	. . . 4 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ (𝐹 “ 𝐾) ∈ 𝐷) → (𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)))
17		f1imacnv 6838	. . . 4 ⊢ ((𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (◡𝐹 “ (𝐹 “ 𝐾)) = 𝐾)
18	16, 17	syl 18	. . 3 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ (𝐹 “ 𝐾) ∈ 𝐷) → (◡𝐹 “ (𝐹 “ 𝐾)) = 𝐾)
19		pm3.22 465	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph))
20	19	3ad2ant1 1151	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph))
21		simpl 488	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → 𝐺 ∈ UHGraph)
22	21	anim1i 627	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐺 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)))
23	22	3adant3 1150	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐺 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)))
24		grimcnv 48791	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → (𝐹 ∈ (𝐺 GraphIso 𝐻) → ◡𝐹 ∈ (𝐻 GraphIso 𝐺)))
25	24	imp 412	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → ◡𝐹 ∈ (𝐻 GraphIso 𝐺))
26	23, 25	syl 18	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → ◡𝐹 ∈ (𝐻 GraphIso 𝐺))
27	26	anim1i 627	. . . 4 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ (𝐹 “ 𝐾) ∈ 𝐷) → (◡𝐹 ∈ (𝐻 GraphIso 𝐺) ∧ (𝐹 “ 𝐾) ∈ 𝐷))
28	5, 4	uhgrimedgi 48793	. . . 4 ⊢ (((𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph) ∧ (◡𝐹 ∈ (𝐻 GraphIso 𝐺) ∧ (𝐹 “ 𝐾) ∈ 𝐷)) → (◡𝐹 “ (𝐹 “ 𝐾)) ∈ 𝐸)
29	20, 27, 28	syl2an2r 698	. . 3 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ (𝐹 “ 𝐾) ∈ 𝐷) → (◡𝐹 “ (𝐹 “ 𝐾)) ∈ 𝐸)
30	18, 29	eqeltrrd 2863	. 2 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ (𝐹 “ 𝐾) ∈ 𝐷) → 𝐾 ∈ 𝐸)
31	7, 30	impbida 813	1 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐾 ∈ 𝐸 ↔ (𝐹 “ 𝐾) ∈ 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3902  ^◡ccnv 5658   “ cima 5662  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7416  Vtxcvtx 29439  Edgcedg 29490  UHGraphcuhgr 29499   GraphIso cgrim 48778

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8831  df-edg 29491  df-uhgr 29501  df-grim 48781

This theorem is used by:  uhgrimprop  48795