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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 2825 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V))) | |
2 | fveq2 6842 | . . . 4 ⊢ (𝑔 = 𝐺 → (2nd ‘𝑔) = (2nd ‘𝐺)) | |
3 | fveq2 6842 | . . . 4 ⊢ (𝑔 = 𝐺 → (.ef‘𝑔) = (.ef‘𝐺)) | |
4 | 1, 2, 3 | ifbieq12d 4514 | . . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))) |
5 | df-iedg 27897 | . . 3 ⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔))) | |
6 | fvex 6855 | . . . 4 ⊢ (2nd ‘𝐺) ∈ V | |
7 | fvex 6855 | . . . 4 ⊢ (.ef‘𝐺) ∈ V | |
8 | 6, 7 | ifex 4536 | . . 3 ⊢ if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) ∈ V |
9 | 4, 5, 8 | fvmpt 6948 | . 2 ⊢ (𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))) |
10 | fvprc 6834 | . . 3 ⊢ (¬ 𝐺 ∈ V → (.ef‘𝐺) = ∅) | |
11 | prcnel 3468 | . . . 4 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V)) | |
12 | 11 | iffalsed 4497 | . . 3 ⊢ (¬ 𝐺 ∈ V → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (.ef‘𝐺)) |
13 | fvprc 6834 | . . 3 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅) | |
14 | 10, 12, 13 | 3eqtr4rd 2787 | . 2 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))) |
15 | 9, 14 | pm2.61i 182 | 1 ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ∅c0 4282 ifcif 4486 × cxp 5631 ‘cfv 6496 2nd c2nd 7919 .efcedgf 27884 iEdgciedg 27895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 df-iedg 27897 |
This theorem is referenced by: opiedgval 27904 funiedgdmge2val 27910 funiedgdm2val 27912 snstriedgval 27936 iedgval0 27938 |
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