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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 2689 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V))) | |
2 | fveq2 6191 | . . . 4 ⊢ (𝑔 = 𝐺 → (2nd ‘𝑔) = (2nd ‘𝐺)) | |
3 | fveq2 6191 | . . . 4 ⊢ (𝑔 = 𝐺 → (.ef‘𝑔) = (.ef‘𝐺)) | |
4 | 1, 2, 3 | ifbieq12d 4113 | . . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))) |
5 | df-iedg 25877 | . . 3 ⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔))) | |
6 | fvex 6201 | . . . 4 ⊢ (2nd ‘𝐺) ∈ V | |
7 | fvex 6201 | . . . 4 ⊢ (.ef‘𝐺) ∈ V | |
8 | 6, 7 | ifex 4156 | . . 3 ⊢ if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) ∈ V |
9 | 4, 5, 8 | fvmpt 6282 | . 2 ⊢ (𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))) |
10 | fvprc 6185 | . . 3 ⊢ (¬ 𝐺 ∈ V → (.ef‘𝐺) = ∅) | |
11 | prcnel 3218 | . . . 4 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V)) | |
12 | 11 | iffalsed 4097 | . . 3 ⊢ (¬ 𝐺 ∈ V → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (.ef‘𝐺)) |
13 | fvprc 6185 | . . 3 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅) | |
14 | 10, 12, 13 | 3eqtr4rd 2667 | . 2 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))) |
15 | 9, 14 | pm2.61i 176 | 1 ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ifcif 4086 × cxp 5112 ‘cfv 5888 2nd c2nd 7167 .efcedgf 25867 iEdgciedg 25875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-iedg 25877 |
This theorem is referenced by: opiedgval 25886 funiedgdmge2val 25892 funiedgdm2val 25894 snstriedgval 25930 iedgval0 25932 |
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