grimedgi - Mathbox for Alexander van der Vekens

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Theorem grimedgi 48290

Description: Graph isomorphisms map edges onto the corresponding edges. (Contributed by AV, 30-Dec-2025.)

**Hypotheses**
Ref	Expression
grimedg.v	⊢ 𝑉 = (Vtx‘𝐺)
grimedg.i	⊢ 𝐼 = (Edg‘𝐺)
grimedg.e	⊢ 𝐸 = (Edg‘𝐻)

**Assertion**
Ref	Expression
grimedgi	⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐾 ∈ 𝐼 → (𝐹 “ 𝐾) ∈ 𝐸))

**Proof of Theorem grimedgi**
Step	Hyp	Ref	Expression
1		grimedg.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		grimedg.i	. . 3 ⊢ 𝐼 = (Edg‘𝐺)
3		grimedg.e	. . 3 ⊢ 𝐸 = (Edg‘𝐻)
4	1, 2, 3	grimedg 48289	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐾 ∈ 𝐼 ↔ ((𝐹 “ 𝐾) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉)))
5		simpl 482	. 2 ⊢ (((𝐹 “ 𝐾) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉) → (𝐹 “ 𝐾) ∈ 𝐸)
6	4, 5	biimtrdi 253	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐾 ∈ 𝐼 → (𝐹 “ 𝐾) ∈ 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ⊆ wss 3903 “ cima 5635 ‘cfv 6500 (class class class)co 7368 Vtxcvtx 29081 Edgcedg 29132 UHGraphcuhgr 29141 GraphIso cgrim 48229

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-map 8777 df-edg 29133 df-uhgr 29143 df-grim 48232

This theorem is referenced by: (None)

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