| Mathbox for BTernaryTau |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > 2cycl2d | Structured version Visualization version GIF version | ||
| Description: Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.) |
| Ref | Expression |
|---|---|
| 2cycl2d.1 | ⊢ 𝑃 = 〈“𝐴𝐵𝐴”〉 |
| 2cycl2d.2 | ⊢ 𝐹 = 〈“𝐽𝐾”〉 |
| 2cycl2d.3 | ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) |
| 2cycl2d.4 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
| 2cycl2d.5 | ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐾))) |
| 2cycl2d.6 | ⊢ 𝑉 = (Vtx‘𝐺) |
| 2cycl2d.7 | ⊢ 𝐼 = (iEdg‘𝐺) |
| 2cycl2d.8 | ⊢ (𝜑 → 𝐽 ≠ 𝐾) |
| Ref | Expression |
|---|---|
| 2cycl2d | ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2cycl2d.1 | . 2 ⊢ 𝑃 = 〈“𝐴𝐵𝐴”〉 | |
| 2 | 2cycl2d.2 | . 2 ⊢ 𝐹 = 〈“𝐽𝐾”〉 | |
| 3 | 2cycl2d.3 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) | |
| 4 | simpl 482 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
| 5 | 3, 4 | jccir 521 | . . 3 ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ∈ 𝑉)) |
| 6 | df-3an 1089 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ∈ 𝑉)) | |
| 7 | 5, 6 | sylibr 234 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)) |
| 8 | 2cycl2d.4 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
| 9 | 8 | necomd 2996 | . . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
| 10 | 8, 9 | jca 511 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐴)) |
| 11 | 2cycl2d.5 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐾))) | |
| 12 | prcom 4732 | . . . . 5 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
| 13 | 12 | sseq1i 4012 | . . . 4 ⊢ ({𝐴, 𝐵} ⊆ (𝐼‘𝐾) ↔ {𝐵, 𝐴} ⊆ (𝐼‘𝐾)) |
| 14 | 13 | anbi2i 623 | . . 3 ⊢ (({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐾)) ↔ ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐴} ⊆ (𝐼‘𝐾))) |
| 15 | 11, 14 | sylib 218 | . 2 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐴} ⊆ (𝐼‘𝐾))) |
| 16 | 2cycl2d.6 | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 17 | 2cycl2d.7 | . 2 ⊢ 𝐼 = (iEdg‘𝐺) | |
| 18 | 2cycl2d.8 | . 2 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
| 19 | eqidd 2738 | . 2 ⊢ (𝜑 → 𝐴 = 𝐴) | |
| 20 | 1, 2, 7, 10, 15, 16, 17, 18, 19 | 2cycld 35143 | 1 ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 {cpr 4628 class class class wbr 5143 ‘cfv 6561 〈“cs2 14880 〈“cs3 14881 Vtxcvtx 29013 iEdgciedg 29014 Cyclesccycls 29805 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 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-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-wlks 29617 df-trls 29710 df-pths 29734 df-cycls 29807 |
| This theorem is referenced by: umgr2cycllem 35145 |
| Copyright terms: Public domain | W3C validator |