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Mirrors > Home > MPE Home > Th. List > iedgval | Structured version Visualization version GIF version |
Description: The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.) |
Ref | Expression |
---|---|
iedgval | ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2900 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V))) | |
2 | fveq2 6656 | . . . 4 ⊢ (𝑔 = 𝐺 → (2nd ‘𝑔) = (2nd ‘𝐺)) | |
3 | fveq2 6656 | . . . 4 ⊢ (𝑔 = 𝐺 → (.ef‘𝑔) = (.ef‘𝐺)) | |
4 | 1, 2, 3 | ifbieq12d 4480 | . . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))) |
5 | df-iedg 26770 | . . 3 ⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔))) | |
6 | fvex 6669 | . . . 4 ⊢ (2nd ‘𝐺) ∈ V | |
7 | fvex 6669 | . . . 4 ⊢ (.ef‘𝐺) ∈ V | |
8 | 6, 7 | ifex 4501 | . . 3 ⊢ if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) ∈ V |
9 | 4, 5, 8 | fvmpt 6754 | . 2 ⊢ (𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))) |
10 | fvprc 6649 | . . 3 ⊢ (¬ 𝐺 ∈ V → (.ef‘𝐺) = ∅) | |
11 | prcnel 3510 | . . . 4 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V)) | |
12 | 11 | iffalsed 4464 | . . 3 ⊢ (¬ 𝐺 ∈ V → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (.ef‘𝐺)) |
13 | fvprc 6649 | . . 3 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅) | |
14 | 10, 12, 13 | 3eqtr4rd 2867 | . 2 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))) |
15 | 9, 14 | pm2.61i 184 | 1 ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3486 ∅c0 4279 ifcif 4453 × cxp 5539 ‘cfv 6341 2nd c2nd 7674 .efcedgf 26760 iEdgciedg 26768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-iota 6300 df-fun 6343 df-fv 6349 df-iedg 26770 |
This theorem is referenced by: opiedgval 26777 funiedgdmge2val 26783 funiedgdm2val 26785 snstriedgval 26809 iedgval0 26811 |
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