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Theorem usgrasscusgra 26011
 Description: An undirected simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
Assertion
Ref Expression
usgrasscusgra ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ∀𝑒 ∈ ran 𝐸𝑓 ∈ ran 𝐹 𝑒 = 𝑓)
Distinct variable groups:   𝑒,𝐸   𝑒,𝐹,𝑓   𝑒,𝑉
Allowed substitution hints:   𝐸(𝑓)   𝑉(𝑓)

Proof of Theorem usgrasscusgra
Dummy variables 𝑎 𝑏 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrarnedg 25913 . . . 4 ((𝑉 USGrph 𝐸𝑒 ∈ ran 𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑒 = {𝑎, 𝑏}))
2 iscusgra0 25986 . . . . . . 7 (𝑉 ComplUSGrph 𝐹 → (𝑉 USGrph 𝐹 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹))
3 simplrr 797 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑒 = {𝑎, 𝑏})) → 𝑏𝑉)
4 sneq 4135 . . . . . . . . . . . . . . 15 (𝑘 = 𝑏 → {𝑘} = {𝑏})
54difeq2d 3690 . . . . . . . . . . . . . 14 (𝑘 = 𝑏 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝑏}))
6 preq2 4213 . . . . . . . . . . . . . . 15 (𝑘 = 𝑏 → {𝑛, 𝑘} = {𝑛, 𝑏})
76eleq1d 2672 . . . . . . . . . . . . . 14 (𝑘 = 𝑏 → ({𝑛, 𝑘} ∈ ran 𝐹 ↔ {𝑛, 𝑏} ∈ ran 𝐹))
85, 7raleqbidv 3129 . . . . . . . . . . . . 13 (𝑘 = 𝑏 → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑏}){𝑛, 𝑏} ∈ ran 𝐹))
98rspcv 3278 . . . . . . . . . . . 12 (𝑏𝑉 → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 → ∀𝑛 ∈ (𝑉 ∖ {𝑏}){𝑛, 𝑏} ∈ ran 𝐹))
103, 9syl 17 . . . . . . . . . . 11 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑒 = {𝑎, 𝑏})) → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 → ∀𝑛 ∈ (𝑉 ∖ {𝑏}){𝑛, 𝑏} ∈ ran 𝐹))
11 simplrl 796 . . . . . . . . . . . . . 14 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → 𝑎𝑉)
12 velsn 4141 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ {𝑏} ↔ 𝑎 = 𝑏)
13 nne 2786 . . . . . . . . . . . . . . . . . . 19 𝑎𝑏𝑎 = 𝑏)
1412, 13bitr4i 266 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ {𝑏} ↔ ¬ 𝑎𝑏)
1514biimpi 205 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {𝑏} → ¬ 𝑎𝑏)
1615a1i 11 . . . . . . . . . . . . . . . 16 ((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎 ∈ {𝑏} → ¬ 𝑎𝑏))
1716con2d 128 . . . . . . . . . . . . . . 15 ((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑏 → ¬ 𝑎 ∈ {𝑏}))
1817imp 444 . . . . . . . . . . . . . 14 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → ¬ 𝑎 ∈ {𝑏})
1911, 18eldifd 3551 . . . . . . . . . . . . 13 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → 𝑎 ∈ (𝑉 ∖ {𝑏}))
2019adantrr 749 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑒 = {𝑎, 𝑏})) → 𝑎 ∈ (𝑉 ∖ {𝑏}))
21 preq1 4212 . . . . . . . . . . . . . 14 (𝑛 = 𝑎 → {𝑛, 𝑏} = {𝑎, 𝑏})
2221eleq1d 2672 . . . . . . . . . . . . 13 (𝑛 = 𝑎 → ({𝑛, 𝑏} ∈ ran 𝐹 ↔ {𝑎, 𝑏} ∈ ran 𝐹))
2322rspcv 3278 . . . . . . . . . . . 12 (𝑎 ∈ (𝑉 ∖ {𝑏}) → (∀𝑛 ∈ (𝑉 ∖ {𝑏}){𝑛, 𝑏} ∈ ran 𝐹 → {𝑎, 𝑏} ∈ ran 𝐹))
2420, 23syl 17 . . . . . . . . . . 11 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑒 = {𝑎, 𝑏})) → (∀𝑛 ∈ (𝑉 ∖ {𝑏}){𝑛, 𝑏} ∈ ran 𝐹 → {𝑎, 𝑏} ∈ ran 𝐹))
25 eleq1 2676 . . . . . . . . . . . . . 14 ({𝑎, 𝑏} = 𝑒 → ({𝑎, 𝑏} ∈ ran 𝐹𝑒 ∈ ran 𝐹))
2625eqcoms 2618 . . . . . . . . . . . . 13 (𝑒 = {𝑎, 𝑏} → ({𝑎, 𝑏} ∈ ran 𝐹𝑒 ∈ ran 𝐹))
27 equid 1926 . . . . . . . . . . . . . 14 𝑒 = 𝑒
28 equequ2 1940 . . . . . . . . . . . . . . 15 (𝑓 = 𝑒 → (𝑒 = 𝑓𝑒 = 𝑒))
2928rspcev 3282 . . . . . . . . . . . . . 14 ((𝑒 ∈ ran 𝐹𝑒 = 𝑒) → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓)
3027, 29mpan2 703 . . . . . . . . . . . . 13 (𝑒 ∈ ran 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓)
3126, 30syl6bi 242 . . . . . . . . . . . 12 (𝑒 = {𝑎, 𝑏} → ({𝑎, 𝑏} ∈ ran 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))
3231ad2antll 761 . . . . . . . . . . 11 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑒 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ ran 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))
3310, 24, 323syld 58 . . . . . . . . . 10 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑒 = {𝑎, 𝑏})) → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))
3433exp31 628 . . . . . . . . 9 (𝑉 USGrph 𝐹 → ((𝑎𝑉𝑏𝑉) → ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))))
3534com24 93 . . . . . . . 8 (𝑉 USGrph 𝐹 → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 → ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → ((𝑎𝑉𝑏𝑉) → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))))
3635imp 444 . . . . . . 7 ((𝑉 USGrph 𝐹 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹) → ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → ((𝑎𝑉𝑏𝑉) → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓)))
372, 36syl 17 . . . . . 6 (𝑉 ComplUSGrph 𝐹 → ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → ((𝑎𝑉𝑏𝑉) → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓)))
3837com13 86 . . . . 5 ((𝑎𝑉𝑏𝑉) → ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → (𝑉 ComplUSGrph 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓)))
3938rexlimivv 3018 . . . 4 (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑒 = {𝑎, 𝑏}) → (𝑉 ComplUSGrph 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))
401, 39syl 17 . . 3 ((𝑉 USGrph 𝐸𝑒 ∈ ran 𝐸) → (𝑉 ComplUSGrph 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))
4140impancom 455 . 2 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (𝑒 ∈ ran 𝐸 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))
4241ralrimiv 2948 1 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ∀𝑒 ∈ ran 𝐸𝑓 ∈ ran 𝐹 𝑒 = 𝑓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   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: (None)
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