Theorem uhgrimedg 48807

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 48806	. . 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 2760	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2760	. . . . . . . . 9 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
10	8, 9	grimf1o 48800	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
11		f1of1 6816	. . . . . . . 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 6834	. . . 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 48804	. . . . . . 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 48806	. . . 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 2861	. 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 3899  ^◡ccnv 5654   “ cima 5658  –1-1→wf1 6530  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7413  Vtxcvtx 29453  Edgcedg 29504  UHGraphcuhgr 29513   GraphIso cgrim 48791

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 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-map 8828  df-edg 29505  df-uhgr 29515  df-grim 48794

This theorem is used by:  uhgrimprop  48808