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Theorem cusgrafilem1 26007
 Description: Lemma 1 for cusgrafi 26010. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
Hypothesis
Ref Expression
cusgrafi.p 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}
Assertion
Ref Expression
cusgrafilem1 ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → 𝑃 ⊆ ran 𝐸)
Distinct variable groups:   𝑁,𝑎,𝑥   𝑉,𝑎,𝑥   𝑥,𝐸
Allowed substitution hints:   𝑃(𝑥,𝑎)   𝐸(𝑎)

Proof of Theorem cusgrafilem1
StepHypRef Expression
1 cusgrarn 25988 . . 3 (𝑉 ComplUSGrph 𝐸 → ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
2 fveq2 6103 . . . . . . . . . 10 (𝑥 = {𝑎, 𝑁} → (#‘𝑥) = (#‘{𝑎, 𝑁}))
32adantl 481 . . . . . . . . 9 ((𝑎𝑁𝑥 = {𝑎, 𝑁}) → (#‘𝑥) = (#‘{𝑎, 𝑁}))
43adantl 481 . . . . . . . 8 ((𝑎𝑉 ∧ (𝑎𝑁𝑥 = {𝑎, 𝑁})) → (#‘𝑥) = (#‘{𝑎, 𝑁}))
54adantl 481 . . . . . . 7 (((𝑁𝑉𝑥 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑥 = {𝑎, 𝑁}))) → (#‘𝑥) = (#‘{𝑎, 𝑁}))
6 simprrl 800 . . . . . . . 8 (((𝑁𝑉𝑥 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑥 = {𝑎, 𝑁}))) → 𝑎𝑁)
7 elex 3185 . . . . . . . . . . 11 (𝑁𝑉𝑁 ∈ V)
8 vex 3176 . . . . . . . . . . 11 𝑎 ∈ V
97, 8jctil 558 . . . . . . . . . 10 (𝑁𝑉 → (𝑎 ∈ V ∧ 𝑁 ∈ V))
109ad2antrr 758 . . . . . . . . 9 (((𝑁𝑉𝑥 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑥 = {𝑎, 𝑁}))) → (𝑎 ∈ V ∧ 𝑁 ∈ V))
11 hashprgOLD 13044 . . . . . . . . 9 ((𝑎 ∈ V ∧ 𝑁 ∈ V) → (𝑎𝑁 ↔ (#‘{𝑎, 𝑁}) = 2))
1210, 11syl 17 . . . . . . . 8 (((𝑁𝑉𝑥 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑥 = {𝑎, 𝑁}))) → (𝑎𝑁 ↔ (#‘{𝑎, 𝑁}) = 2))
136, 12mpbid 221 . . . . . . 7 (((𝑁𝑉𝑥 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑥 = {𝑎, 𝑁}))) → (#‘{𝑎, 𝑁}) = 2)
145, 13eqtrd 2644 . . . . . 6 (((𝑁𝑉𝑥 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑥 = {𝑎, 𝑁}))) → (#‘𝑥) = 2)
1514rexlimdvaa 3014 . . . . 5 ((𝑁𝑉𝑥 ∈ 𝒫 𝑉) → (∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁}) → (#‘𝑥) = 2))
1615ss2rabdv 3646 . . . 4 (𝑁𝑉 → {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
17 cusgrafi.p . . . . . 6 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}
1817a1i 11 . . . . 5 (ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})})
19 id 22 . . . . 5 (ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
2018, 19sseq12d 3597 . . . 4 (ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (𝑃 ⊆ ran 𝐸 ↔ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
2116, 20syl5ibr 235 . . 3 (ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (𝑁𝑉𝑃 ⊆ ran 𝐸))
221, 21syl 17 . 2 (𝑉 ComplUSGrph 𝐸 → (𝑁𝑉𝑃 ⊆ ran 𝐸))
2322imp 444 1 ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → 𝑃 ⊆ ran 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  {cpr 4127   class class class wbr 4583  ran crn 5039  ‘cfv 5804  2c2 10947  #chash 12979   ComplUSGrph ccusgra 25947 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-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-usgra 25862  df-cusgra 25950 This theorem is referenced by:  cusgrafi  26010
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