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Mirrors > Home > MPE Home > Th. List > upgr1wlkdlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for upgr1wlkd 30176. (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 4739 | . . . . . . 7 ⊢ (𝑌 = 𝑋 → {𝑋, 𝑌} = {𝑋, 𝑋}) | |
3 | 2 | eqeq2d 2746 | . . . . . 6 ⊢ (𝑌 = 𝑋 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋})) |
4 | 3 | eqcoms 2743 | . . . . 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 4644 | . . . . . . 7 ⊢ {𝑋} = {𝑋, 𝑋} | |
7 | 5, 6 | eqtr4di 2793 | . . . . . 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 1537 ∈ wcel 2106 {csn 4631 {cpr 4633 ‘cfv 6563 〈“cs1 14630 〈“cs2 14877 Vtxcvtx 29028 iEdgciedg 29029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-sn 4632 df-pr 4634 |
This theorem is referenced by: upgr1wlkd 30176 upgr1trld 30177 upgr1pthd 30178 upgr1pthond 30179 |
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