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Theorem cusgrfilem1 29544

Description: Lemma 1 for cusgrfi 29547. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)

**Hypotheses**
Ref	Expression
cusgrfi.v	⊢ 𝑉 = (Vtx‘𝐺)
cusgrfi.p	⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}

**Assertion**
Ref	Expression
cusgrfilem1	⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑃 ⊆ (Edg‘𝐺))

Distinct variable groups: 𝑥,𝐺 𝑁,𝑎,𝑥 𝑉,𝑎,𝑥 Allowed substitution hints: 𝑃(𝑥,𝑎) 𝐺(𝑎)

**Proof of Theorem cusgrfilem1**
Step	Hyp	Ref	Expression
1		cusgrfi.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2737	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	cusgredg 29512	. . 3 ⊢ (𝐺 ∈ ComplUSGraph → (Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
4		fveq2 6832	. . . . . . . . 9 ⊢ (𝑥 = {𝑎, 𝑁} → (♯‘𝑥) = (♯‘{𝑎, 𝑁}))
5	4	ad2antlr 728	. . . . . . . 8 ⊢ (((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ∧ (𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉))) → (♯‘𝑥) = (♯‘{𝑎, 𝑁}))
6		hashprg 14346	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝑎 ≠ 𝑁 ↔ (♯‘{𝑎, 𝑁}) = 2))
7	6	adantrr 718	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉)) → (𝑎 ≠ 𝑁 ↔ (♯‘{𝑎, 𝑁}) = 2))
8	7	biimpcd 249	. . . . . . . . . 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 406	. . . . . . . 8 ⊢ (((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ∧ (𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉))) → (♯‘{𝑎, 𝑁}) = 2)
11	5, 10	eqtrd 2772	. . . . . . 7 ⊢ (((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ∧ (𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉))) → (♯‘𝑥) = 2)
12	11	an13s 652	. . . . . 6 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}))) → (♯‘𝑥) = 2)
13	12	rexlimdvaa 3140	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉) → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) → (♯‘𝑥) = 2))
14	13	ss2rabdv 4016	. . . 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 3956	. . . 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	imbitrrid 246	. . 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 406	1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑃 ⊆ (Edg‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 {crab 3390 ⊆ wss 3890 𝒫 cpw 4542 {cpr 4570 ‘cfv 6490 2c2 12225 ♯chash 14281 Vtxcvtx 29084 Edgcedg 29135 ComplUSGraphccusgr 29498

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-er 8634 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-dju 9814 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-n0 12427 df-xnn0 12500 df-z 12514 df-uz 12778 df-fz 13451 df-hash 14282 df-edg 29136 df-upgr 29170 df-umgr 29171 df-usgr 29239 df-nbgr 29421 df-uvtx 29474 df-cplgr 29499 df-cusgr 29500

This theorem is referenced by: cusgrfi 29547

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