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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.) |
Ref | Expression |
---|---|
vtxvalsnop.b | ⊢ 𝐵 ∈ V |
vtxvalsnop.g | ⊢ 𝐺 = {〈𝐵, 𝐵〉} |
Ref | Expression |
---|---|
iedgvalsnop | ⊢ (iEdg‘𝐺) = {𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ 𝐵 = 𝐵 | |
2 | vtxvalsnop.b | . . . 4 ⊢ 𝐵 ∈ V | |
3 | vtxvalsnop.g | . . . 4 ⊢ 𝐺 = {〈𝐵, 𝐵〉} | |
4 | 2, 2, 3 | funsneqopsn 6322 | . . 3 ⊢ (𝐵 = 𝐵 → 𝐺 = 〈{𝐵}, {𝐵}〉) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ 𝐺 = 〈{𝐵}, {𝐵}〉 |
6 | fveq2 6103 | . . 3 ⊢ (𝐺 = 〈{𝐵}, {𝐵}〉 → (iEdg‘𝐺) = (iEdg‘〈{𝐵}, {𝐵}〉)) | |
7 | opex 4859 | . . . 4 ⊢ 〈{𝐵}, {𝐵}〉 ∈ V | |
8 | iedgval 25678 | . . . . 5 ⊢ (〈{𝐵}, {𝐵}〉 ∈ V → (iEdg‘〈{𝐵}, {𝐵}〉) = if(〈{𝐵}, {𝐵}〉 ∈ (V × V), (2nd ‘〈{𝐵}, {𝐵}〉), (.ef‘〈{𝐵}, {𝐵}〉))) | |
9 | snex 4835 | . . . . . . . 8 ⊢ {𝐵} ∈ V | |
10 | 9, 9 | opelvv 5088 | . . . . . . 7 ⊢ 〈{𝐵}, {𝐵}〉 ∈ (V × V) |
11 | 10 | iftruei 4043 | . . . . . 6 ⊢ if(〈{𝐵}, {𝐵}〉 ∈ (V × V), (2nd ‘〈{𝐵}, {𝐵}〉), (.ef‘〈{𝐵}, {𝐵}〉)) = (2nd ‘〈{𝐵}, {𝐵}〉) |
12 | 9, 9 | op2nd 7068 | . . . . . 6 ⊢ (2nd ‘〈{𝐵}, {𝐵}〉) = {𝐵} |
13 | 11, 12 | eqtri 2632 | . . . . 5 ⊢ if(〈{𝐵}, {𝐵}〉 ∈ (V × V), (2nd ‘〈{𝐵}, {𝐵}〉), (.ef‘〈{𝐵}, {𝐵}〉)) = {𝐵} |
14 | 8, 13 | syl6eq 2660 | . . . 4 ⊢ (〈{𝐵}, {𝐵}〉 ∈ V → (iEdg‘〈{𝐵}, {𝐵}〉) = {𝐵}) |
15 | 7, 14 | ax-mp 5 | . . 3 ⊢ (iEdg‘〈{𝐵}, {𝐵}〉) = {𝐵} |
16 | 6, 15 | syl6eq 2660 | . 2 ⊢ (𝐺 = 〈{𝐵}, {𝐵}〉 → (iEdg‘𝐺) = {𝐵}) |
17 | 5, 16 | ax-mp 5 | 1 ⊢ (iEdg‘𝐺) = {𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 {csn 4125 〈cop 4131 × cxp 5036 ‘cfv 5804 2nd c2nd 7058 .efcedgf 25667 iEdgciedg 25674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-2nd 7060 df-iedg 25676 |
This theorem is referenced by: iedgval3sn 25719 |
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