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| Mirrors > Home > MPE Home > Th. List > iedgvalsnop | Structured version Visualization version GIF version | ||
| Description: Degenerated case 2 for edges: The set of indexed edges of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 15-Jul-2022.) (Avoid depending on this detail.) |
| Ref | Expression |
|---|---|
| vtxvalsnop.b | ⊢ 𝐵 ∈ V |
| vtxvalsnop.g | ⊢ 𝐺 = {〈𝐵, 𝐵〉} |
| Ref | Expression |
|---|---|
| iedgvalsnop | ⊢ (iEdg‘𝐺) = {𝐵} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vtxvalsnop.g | . . 3 ⊢ 𝐺 = {〈𝐵, 𝐵〉} | |
| 2 | 1 | fveq2i 6909 | . 2 ⊢ (iEdg‘𝐺) = (iEdg‘{〈𝐵, 𝐵〉}) |
| 3 | vtxvalsnop.b | . . . 4 ⊢ 𝐵 ∈ V | |
| 4 | 3 | snopeqopsnid 5514 | . . 3 ⊢ {〈𝐵, 𝐵〉} = 〈{𝐵}, {𝐵}〉 |
| 5 | 4 | fveq2i 6909 | . 2 ⊢ (iEdg‘{〈𝐵, 𝐵〉}) = (iEdg‘〈{𝐵}, {𝐵}〉) |
| 6 | snex 5436 | . . 3 ⊢ {𝐵} ∈ V | |
| 7 | 6, 6 | opiedgfvi 29027 | . 2 ⊢ (iEdg‘〈{𝐵}, {𝐵}〉) = {𝐵} |
| 8 | 2, 5, 7 | 3eqtri 2769 | 1 ⊢ (iEdg‘𝐺) = {𝐵} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ‘cfv 6561 iEdgciedg 29014 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-2nd 8015 df-iedg 29016 |
| This theorem is referenced by: iedgval3sn 29061 |
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