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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 28817 | . 2 ⊢ (Edg‘⟨𝑉, 𝐸⟩) = ran (iEdg‘⟨𝑉, 𝐸⟩) | |
2 | opiedgfv 28775 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) | |
3 | 2 | rneqd 5931 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ran (iEdg‘⟨𝑉, 𝐸⟩) = ran 𝐸) |
4 | 1, 3 | eqtrid 2778 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⟨cop 4629 ran crn 5670 ‘cfv 6537 iEdgciedg 28765 Edgcedg 28815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fv 6545 df-2nd 7975 df-iedg 28767 df-edg 28816 |
This theorem is referenced by: edgov 28820 cusgrsize 29220 uspgrloopedg 29284 uspgrsprfo 47095 |
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