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| Mirrors > Home > MPE Home > Th. List > uspgrloopiedg | Structured version Visualization version GIF version | ||
| Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29266) is a singleton of a singleton. (Contributed by AV, 21-Feb-2021.) | 
| Ref | Expression | 
|---|---|
| uspgrloopvtx.g | ⊢ 𝐺 = 〈𝑉, {〈𝐴, {𝑁}〉}〉 | 
| Ref | Expression | 
|---|---|
| uspgrloopiedg | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (iEdg‘𝐺) = {〈𝐴, {𝑁}〉}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uspgrloopvtx.g | . . 3 ⊢ 𝐺 = 〈𝑉, {〈𝐴, {𝑁}〉}〉 | |
| 2 | 1 | fveq2i 6909 | . 2 ⊢ (iEdg‘𝐺) = (iEdg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) | 
| 3 | snex 5436 | . . . 4 ⊢ {〈𝐴, {𝑁}〉} ∈ V | |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑋 → {〈𝐴, {𝑁}〉} ∈ V) | 
| 5 | opiedgfv 29024 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ {〈𝐴, {𝑁}〉} ∈ V) → (iEdg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = {〈𝐴, {𝑁}〉}) | |
| 6 | 4, 5 | sylan2 593 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (iEdg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = {〈𝐴, {𝑁}〉}) | 
| 7 | 2, 6 | eqtrid 2789 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (iEdg‘𝐺) = {〈𝐴, {𝑁}〉}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = 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: uspgrloopnb0 29537 uspgrloopvd2 29538 | 
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