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| Mirrors > Home > MPE Home > Th. List > Mathboxes > grimf1o | Structured version Visualization version GIF version | ||
| Description: An isomorphism of graphs is a bijection between their vertices. (Contributed by AV, 29-Apr-2025.) | 
| Ref | Expression | 
|---|---|
| grimprop.v | ⊢ 𝑉 = (Vtx‘𝐺) | 
| grimprop.w | ⊢ 𝑊 = (Vtx‘𝐻) | 
| Ref | Expression | 
|---|---|
| grimf1o | ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | grimprop.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 2 | grimprop.w | . . 3 ⊢ 𝑊 = (Vtx‘𝐻) | |
| 3 | eqid 2736 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 4 | eqid 2736 | . . 3 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻) | |
| 5 | 1, 2, 3, 4 | grimprop 47874 | . 2 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))))) | 
| 6 | 5 | simpld 494 | 1 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ∀wral 3060 dom cdm 5684 “ cima 5687 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 Vtxcvtx 29014 iEdgciedg 29015 GraphIso cgrim 47866 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-grim 47869 | 
| This theorem is referenced by: gricen 47899 clnbgrgrimlem 47906 clnbgrgrim 47907 grimgrtri 47921 uhgrimgrlim 47959 | 
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