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| Description: Lemma 1 for upgr1wlkd 30167. (Contributed by AV, 22-Jan-2021.) | 
| Ref | Expression | 
|---|---|
| upgr1wlkd.p | ⊢ 𝑃 = 〈“𝑋𝑌”〉 | 
| upgr1wlkd.f | ⊢ 𝐹 = 〈“𝐽”〉 | 
| upgr1wlkd.x | ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) | 
| upgr1wlkd.y | ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) | 
| upgr1wlkd.j | ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) | 
| Ref | Expression | 
|---|---|
| upgr1wlkdlem1 | ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | upgr1wlkd.j | . . 3 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) | |
| 2 | preq2 4733 | . . . . . . 7 ⊢ (𝑌 = 𝑋 → {𝑋, 𝑌} = {𝑋, 𝑋}) | |
| 3 | 2 | eqeq2d 2747 | . . . . . 6 ⊢ (𝑌 = 𝑋 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋})) | 
| 4 | 3 | eqcoms 2744 | . . . . 5 ⊢ (𝑋 = 𝑌 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋})) | 
| 5 | simpl 482 | . . . . . . 7 ⊢ ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}) | |
| 6 | dfsn2 4638 | . . . . . . 7 ⊢ {𝑋} = {𝑋, 𝑋} | |
| 7 | 5, 6 | eqtr4di 2794 | . . . . . 6 ⊢ ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋}) | 
| 8 | 7 | ex 412 | . . . . 5 ⊢ (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋})) | 
| 9 | 4, 8 | biimtrdi 253 | . . . 4 ⊢ (𝑋 = 𝑌 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋}))) | 
| 10 | 9 | com13 88 | . . 3 ⊢ (𝜑 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → (𝑋 = 𝑌 → ((iEdg‘𝐺)‘𝐽) = {𝑋}))) | 
| 11 | 1, 10 | mpd 15 | . 2 ⊢ (𝜑 → (𝑋 = 𝑌 → ((iEdg‘𝐺)‘𝐽) = {𝑋})) | 
| 12 | 11 | imp 406 | 1 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {csn 4625 {cpr 4627 ‘cfv 6560 〈“cs1 14634 〈“cs2 14881 Vtxcvtx 29014 iEdgciedg 29015 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 df-sn 4626 df-pr 4628 | 
| This theorem is referenced by: upgr1wlkd 30167 upgr1trld 30168 upgr1pthd 30169 upgr1pthond 30170 | 
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