MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cusgrfilem2 Structured version   Visualization version   GIF version

Theorem cusgrfilem2 29489
Description: Lemma 2 for cusgrfi 29491. (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 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}
cusgrfi.f 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})
Assertion
Ref Expression
cusgrfilem2 (𝑁𝑉𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃)
Distinct variable groups:   𝑥,𝐺   𝑁,𝑎,𝑥   𝑉,𝑎,𝑥   𝑥,𝑃
Allowed substitution hints:   𝑃(𝑎)   𝐹(𝑥,𝑎)   𝐺(𝑎)

Proof of Theorem cusgrfilem2
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifi 4141 . . . . 5 (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥𝑉)
2 id 22 . . . . 5 (𝑁𝑉𝑁𝑉)
3 prelpwi 5458 . . . . 5 ((𝑥𝑉𝑁𝑉) → {𝑥, 𝑁} ∈ 𝒫 𝑉)
41, 2, 3syl2anr 597 . . . 4 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} ∈ 𝒫 𝑉)
51adantl 481 . . . . 5 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥𝑉)
6 eldifsni 4795 . . . . . . 7 (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥𝑁)
76adantl 481 . . . . . 6 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥𝑁)
8 eqidd 2736 . . . . . 6 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} = {𝑥, 𝑁})
97, 8jca 511 . . . . 5 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁}))
10 id 22 . . . . . 6 (𝑥𝑉𝑥𝑉)
11 neeq1 3001 . . . . . . . 8 (𝑎 = 𝑥 → (𝑎𝑁𝑥𝑁))
12 preq1 4738 . . . . . . . . 9 (𝑎 = 𝑥 → {𝑎, 𝑁} = {𝑥, 𝑁})
1312eqeq2d 2746 . . . . . . . 8 (𝑎 = 𝑥 → ({𝑥, 𝑁} = {𝑎, 𝑁} ↔ {𝑥, 𝑁} = {𝑥, 𝑁}))
1411, 13anbi12d 632 . . . . . . 7 (𝑎 = 𝑥 → ((𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁})))
1514adantl 481 . . . . . 6 ((𝑥𝑉𝑎 = 𝑥) → ((𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁})))
1610, 15rspcedv 3615 . . . . 5 (𝑥𝑉 → ((𝑥𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁}) → ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
175, 9, 16sylc 65 . . . 4 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}))
18 cusgrfi.p . . . . . 6 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}
1918eleq2i 2831 . . . . 5 ({𝑥, 𝑁} ∈ 𝑃 ↔ {𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})})
20 eqeq1 2739 . . . . . . . 8 (𝑣 = {𝑥, 𝑁} → (𝑣 = {𝑎, 𝑁} ↔ {𝑥, 𝑁} = {𝑎, 𝑁}))
2120anbi2d 630 . . . . . . 7 (𝑣 = {𝑥, 𝑁} → ((𝑎𝑁𝑣 = {𝑎, 𝑁}) ↔ (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
2221rexbidv 3177 . . . . . 6 (𝑣 = {𝑥, 𝑁} → (∃𝑎𝑉 (𝑎𝑁𝑣 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
23 eqeq1 2739 . . . . . . . . 9 (𝑥 = 𝑣 → (𝑥 = {𝑎, 𝑁} ↔ 𝑣 = {𝑎, 𝑁}))
2423anbi2d 630 . . . . . . . 8 (𝑥 = 𝑣 → ((𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ (𝑎𝑁𝑣 = {𝑎, 𝑁})))
2524rexbidv 3177 . . . . . . 7 (𝑥 = 𝑣 → (∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁𝑣 = {𝑎, 𝑁})))
2625cbvrabv 3444 . . . . . 6 {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})} = {𝑣 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑣 = {𝑎, 𝑁})}
2722, 26elrab2 3698 . . . . 5 ({𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})} ↔ ({𝑥, 𝑁} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
2819, 27bitri 275 . . . 4 ({𝑥, 𝑁} ∈ 𝑃 ↔ ({𝑥, 𝑁} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
294, 17, 28sylanbrc 583 . . 3 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} ∈ 𝑃)
3029ralrimiva 3144 . 2 (𝑁𝑉 → ∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃)
31 simpl 482 . . . . . . . . . . 11 ((𝑎𝑁𝑒 = {𝑎, 𝑁}) → 𝑎𝑁)
3231anim2i 617 . . . . . . . . . 10 ((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) → (𝑎𝑉𝑎𝑁))
3332adantl 481 . . . . . . . . 9 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → (𝑎𝑉𝑎𝑁))
34 eldifsn 4791 . . . . . . . . 9 (𝑎 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑎𝑉𝑎𝑁))
3533, 34sylibr 234 . . . . . . . 8 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → 𝑎 ∈ (𝑉 ∖ {𝑁}))
36 eqeq1 2739 . . . . . . . . . . . . . 14 (𝑒 = {𝑎, 𝑁} → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
3736adantl 481 . . . . . . . . . . . . 13 ((𝑎𝑁𝑒 = {𝑎, 𝑁}) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
3837ad2antlr 727 . . . . . . . . . . . 12 (((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
39 vex 3482 . . . . . . . . . . . . . 14 𝑎 ∈ V
40 vex 3482 . . . . . . . . . . . . . 14 𝑥 ∈ V
4139, 40preqr1 4853 . . . . . . . . . . . . 13 ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑎 = 𝑥)
4241equcomd 2016 . . . . . . . . . . . 12 ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑥 = 𝑎)
4338, 42biimtrdi 253 . . . . . . . . . . 11 (((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
4443adantll 714 . . . . . . . . . 10 ((((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
4512equcoms 2017 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → {𝑎, 𝑁} = {𝑥, 𝑁})
4645eqeq2d 2746 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑒 = {𝑎, 𝑁} ↔ 𝑒 = {𝑥, 𝑁}))
4746biimpcd 249 . . . . . . . . . . . . 13 (𝑒 = {𝑎, 𝑁} → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
4847adantl 481 . . . . . . . . . . . 12 ((𝑎𝑁𝑒 = {𝑎, 𝑁}) → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
4948adantl 481 . . . . . . . . . . 11 ((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
5049ad2antlr 727 . . . . . . . . . 10 ((((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
5144, 50impbid 212 . . . . . . . . 9 ((((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
5251ralrimiva 3144 . . . . . . . 8 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
5335, 52jca 511 . . . . . . 7 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
5453ex 412 . . . . . 6 ((𝑁𝑉𝑒 ∈ 𝒫 𝑉) → ((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))))
5554reximdv2 3162 . . . . 5 ((𝑁𝑉𝑒 ∈ 𝒫 𝑉) → (∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁}) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
5655expimpd 453 . . . 4 (𝑁𝑉 → ((𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
57 eqeq1 2739 . . . . . . 7 (𝑥 = 𝑒 → (𝑥 = {𝑎, 𝑁} ↔ 𝑒 = {𝑎, 𝑁}))
5857anbi2d 630 . . . . . 6 (𝑥 = 𝑒 → ((𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ (𝑎𝑁𝑒 = {𝑎, 𝑁})))
5958rexbidv 3177 . . . . 5 (𝑥 = 𝑒 → (∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})))
6059, 18elrab2 3698 . . . 4 (𝑒𝑃 ↔ (𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})))
61 reu6 3735 . . . 4 (∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁} ↔ ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
6256, 60, 613imtr4g 296 . . 3 (𝑁𝑉 → (𝑒𝑃 → ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
6362ralrimiv 3143 . 2 (𝑁𝑉 → ∀𝑒𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁})
64 cusgrfi.f . . 3 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})
6564f1ompt 7131 . 2 (𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃 ↔ (∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃 ∧ ∀𝑒𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
6630, 63, 65sylanbrc 583 1 (𝑁𝑉𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  ∃!wreu 3376  {crab 3433  cdif 3960  𝒫 cpw 4605  {csn 4631  {cpr 4633  cmpt 5231  1-1-ontowf1o 6562  cfv 6563  Vtxcvtx 29028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by:  cusgrfilem3  29490
  Copyright terms: Public domain W3C validator