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Mirrors > Home > MPE Home > Th. List > upgr1wlkdlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for upgr1wlkd 27676. (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 4544 | . . . . . . 7 ⊢ (𝑌 = 𝑋 → {𝑋, 𝑌} = {𝑋, 𝑋}) | |
3 | 2 | eqeq2d 2789 | . . . . . 6 ⊢ (𝑌 = 𝑋 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋})) |
4 | 3 | eqcoms 2787 | . . . . 5 ⊢ (𝑋 = 𝑌 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋})) |
5 | simpl 475 | . . . . . . 7 ⊢ ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}) | |
6 | dfsn2 4454 | . . . . . . 7 ⊢ {𝑋} = {𝑋, 𝑋} | |
7 | 5, 6 | syl6eqr 2833 | . . . . . 6 ⊢ ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋}) |
8 | 7 | ex 405 | . . . . 5 ⊢ (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋})) |
9 | 4, 8 | syl6bi 245 | . . . 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 398 | 1 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 {csn 4441 {cpr 4443 ‘cfv 6188 〈“cs1 13758 〈“cs2 14065 Vtxcvtx 26484 iEdgciedg 26485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2751 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-v 3418 df-un 3835 df-sn 4442 df-pr 4444 |
This theorem is referenced by: upgr1wlkd 27676 upgr1trld 27677 upgr1pthd 27678 upgr1pthond 27679 |
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