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Mirrors > Home > MPE Home > Th. List > edgopval | Structured version Visualization version GIF version |
Description: The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) |
Ref | Expression |
---|---|
edgopval | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘〈𝑉, 𝐸〉) = ran 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | edgval 26284 | . 2 ⊢ (Edg‘〈𝑉, 𝐸〉) = ran (iEdg‘〈𝑉, 𝐸〉) | |
2 | opiedgfv 26242 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
3 | 2 | rneqd 5556 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ran (iEdg‘〈𝑉, 𝐸〉) = ran 𝐸) |
4 | 1, 3 | syl5eq 2845 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘〈𝑉, 𝐸〉) = ran 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 〈cop 4374 ran crn 5313 ‘cfv 6101 iEdgciedg 26232 Edgcedg 26282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-iota 6064 df-fun 6103 df-fv 6109 df-2nd 7402 df-iedg 26234 df-edg 26283 |
This theorem is referenced by: edgov 26287 cusgrsize 26704 uspgrloopedg 26768 uspgrsprfo 42555 |
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