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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 2772 | . . 3 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
2 | 1 | isfusgr 26797 | . 2 ⊢ (〈𝑉, 𝐸〉 ∈ FinUSGraph ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ (Vtx‘〈𝑉, 𝐸〉) ∈ Fin)) |
3 | opvtxfv 26486 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
4 | 3 | eleq1d 2844 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((Vtx‘〈𝑉, 𝐸〉) ∈ Fin ↔ 𝑉 ∈ Fin)) |
5 | 4 | anbi2d 619 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((〈𝑉, 𝐸〉 ∈ USGraph ∧ (Vtx‘〈𝑉, 𝐸〉) ∈ Fin) ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ 𝑉 ∈ Fin))) |
6 | 2, 5 | syl5bb 275 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (〈𝑉, 𝐸〉 ∈ FinUSGraph ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ 𝑉 ∈ Fin))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∈ wcel 2050 〈cop 4441 ‘cfv 6182 Fincfn 8300 Vtxcvtx 26478 USGraphcusgr 26631 FinUSGraphcfusgr 26795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5306 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-iota 6146 df-fun 6184 df-fv 6190 df-1st 7495 df-vtx 26480 df-fusgr 26796 |
This theorem is referenced by: fusgrfis 26809 |
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