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Mirrors > Home > MPE Home > Th. List > edgaopval | 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 |
---|---|
edgaopval | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘〈𝑉, 𝐸〉) = ran 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 4859 | . . . 4 ⊢ 〈𝑉, 𝐸〉 ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 〈𝑉, 𝐸〉 ∈ V) |
3 | edgaval 25794 | . . 3 ⊢ (〈𝑉, 𝐸〉 ∈ V → (Edg‘〈𝑉, 𝐸〉) = ran (iEdg‘〈𝑉, 𝐸〉)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘〈𝑉, 𝐸〉) = ran (iEdg‘〈𝑉, 𝐸〉)) |
5 | opiedgfv 25684 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
6 | 5 | rneqd 5274 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ran (iEdg‘〈𝑉, 𝐸〉) = ran 𝐸) |
7 | 4, 6 | eqtrd 2644 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘〈𝑉, 𝐸〉) = ran 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ran crn 5039 ‘cfv 5804 iEdgciedg 25674 Edgcedga 25792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-2nd 7060 df-iedg 25676 df-edga 25793 |
This theorem is referenced by: edgaov 25796 cusgrsize 40670 uspgrloopedg 40734 |
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