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Mirrors > Home > MPE Home > Th. List > Mathboxes > grimprop | Structured version Visualization version GIF version |
Description: Properties of an isomorphism of graphs. (Contributed by AV, 29-Apr-2025.) |
Ref | Expression |
---|---|
grimprop.v | ⊢ 𝑉 = (Vtx‘𝐺) |
grimprop.w | ⊢ 𝑊 = (Vtx‘𝐻) |
grimprop.e | ⊢ 𝐸 = (iEdg‘𝐺) |
grimprop.d | ⊢ 𝐷 = (iEdg‘𝐻) |
Ref | Expression |
---|---|
grimprop | ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grimdmrel 47804 | . . . . 5 ⊢ Rel dom GraphIso | |
2 | 1 | ovrcl 7472 | . . . 4 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V)) |
3 | 2 | simpld 494 | . . 3 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐺 ∈ V) |
4 | 2 | simprd 495 | . . 3 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐻 ∈ V) |
5 | id 22 | . . 3 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹 ∈ (𝐺 GraphIso 𝐻)) | |
6 | 3, 4, 5 | 3jca 1127 | . 2 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻))) |
7 | grimprop.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | grimprop.w | . . . 4 ⊢ 𝑊 = (Vtx‘𝐻) | |
9 | grimprop.e | . . . 4 ⊢ 𝐸 = (iEdg‘𝐺) | |
10 | grimprop.d | . . . 4 ⊢ 𝐷 = (iEdg‘𝐻) | |
11 | 7, 8, 9, 10 | isgrim 47806 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖)))))) |
12 | 11 | biimpd 229 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖)))))) |
13 | 6, 12 | mpcom 38 | 1 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 dom cdm 5689 “ cima 5692 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 Vtxcvtx 29028 iEdgciedg 29029 GraphIso cgrim 47799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-grim 47802 |
This theorem is referenced by: grimf1o 47808 grimuhgr 47816 grimcnv 47817 grimco 47818 uhgrimisgrgric 47837 clnbgrgrimlem 47839 clnbgrgrim 47840 grimedg 47841 |
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