Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > cusgrfilem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for cusgrfi 40674. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
Ref | Expression |
---|---|
cusgrfi.v | ⊢ 𝑉 = (Vtx‘𝐺) |
cusgrfi.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} |
Ref | Expression |
---|---|
cusgrfilem1 | ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑃 ⊆ (Edg‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cusgrfi.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | eqid 2610 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | cusgredg 40646 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → (Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}) |
4 | fveq2 6103 | . . . . . . . . 9 ⊢ (𝑥 = {𝑎, 𝑁} → (#‘𝑥) = (#‘{𝑎, 𝑁})) | |
5 | 4 | ad2antlr 759 | . . . . . . . 8 ⊢ (((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ∧ (𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉))) → (#‘𝑥) = (#‘{𝑎, 𝑁})) |
6 | hashprg 13043 | . . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝑎 ≠ 𝑁 ↔ (#‘{𝑎, 𝑁}) = 2)) | |
7 | 6 | adantrr 749 | . . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉)) → (𝑎 ≠ 𝑁 ↔ (#‘{𝑎, 𝑁}) = 2)) |
8 | 7 | biimpcd 238 | . . . . . . . . . 10 ⊢ (𝑎 ≠ 𝑁 → ((𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉)) → (#‘{𝑎, 𝑁}) = 2)) |
9 | 8 | adantr 480 | . . . . . . . . 9 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) → ((𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉)) → (#‘{𝑎, 𝑁}) = 2)) |
10 | 9 | imp 444 | . . . . . . . 8 ⊢ (((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ∧ (𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉))) → (#‘{𝑎, 𝑁}) = 2) |
11 | 5, 10 | eqtrd 2644 | . . . . . . 7 ⊢ (((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ∧ (𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉))) → (#‘𝑥) = 2) |
12 | 11 | an13s 841 | . . . . . 6 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}))) → (#‘𝑥) = 2) |
13 | 12 | rexlimdvaa 3014 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉) → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) → (#‘𝑥) = 2)) |
14 | 13 | ss2rabdv 3646 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}) |
15 | cusgrfi.p | . . . . . 6 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} | |
16 | 15 | a1i 11 | . . . . 5 ⊢ ((Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}) |
17 | id 22 | . . . . 5 ⊢ ((Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}) | |
18 | 16, 17 | sseq12d 3597 | . . . 4 ⊢ ((Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (𝑃 ⊆ (Edg‘𝐺) ↔ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})) |
19 | 14, 18 | syl5ibr 235 | . . 3 ⊢ ((Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (𝑁 ∈ 𝑉 → 𝑃 ⊆ (Edg‘𝐺))) |
20 | 3, 19 | syl 17 | . 2 ⊢ (𝐺 ∈ ComplUSGraph → (𝑁 ∈ 𝑉 → 𝑃 ⊆ (Edg‘𝐺))) |
21 | 20 | imp 444 | 1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑃 ⊆ (Edg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 {crab 2900 ⊆ wss 3540 𝒫 cpw 4108 {cpr 4127 ‘cfv 5804 2c2 10947 #chash 12979 Vtxcvtx 25673 Edgcedga 25792 ComplUSGraphccusgr 40553 |
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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-upgr 25749 df-umgr 25750 df-edga 25793 df-usgr 40381 df-nbgr 40554 df-uvtxa 40556 df-cplgr 40557 df-cusgr 40558 |
This theorem is referenced by: cusgrfi 40674 |
Copyright terms: Public domain | W3C validator |