![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > opfusgr | Structured version Visualization version GIF version |
Description: A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020.) |
Ref | Expression |
---|---|
opfusgr | ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (〈𝑉, 𝐸〉 ∈ FinUSGraph ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ 𝑉 ∈ Fin))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . . 3 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
2 | 1 | isfusgr 29118 | . 2 ⊢ (〈𝑉, 𝐸〉 ∈ FinUSGraph ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ (Vtx‘〈𝑉, 𝐸〉) ∈ Fin)) |
3 | opvtxfv 28804 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
4 | 3 | eleq1d 2813 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((Vtx‘〈𝑉, 𝐸〉) ∈ Fin ↔ 𝑉 ∈ Fin)) |
5 | 4 | anbi2d 628 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((〈𝑉, 𝐸〉 ∈ USGraph ∧ (Vtx‘〈𝑉, 𝐸〉) ∈ Fin) ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ 𝑉 ∈ Fin))) |
6 | 2, 5 | bitrid 283 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (〈𝑉, 𝐸〉 ∈ FinUSGraph ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ 𝑉 ∈ Fin))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2099 〈cop 4630 ‘cfv 6542 Fincfn 8955 Vtxcvtx 28796 USGraphcusgr 28949 FinUSGraphcfusgr 29116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fv 6550 df-1st 7987 df-vtx 28798 df-fusgr 29117 |
This theorem is referenced by: fusgrfis 29130 |
Copyright terms: Public domain | W3C validator |