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Theorem cusgrfilem2 29474
Description: Lemma 2 for cusgrfi 29476. (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 4131 . . . . 5 (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥𝑉)
2 id 22 . . . . 5 (𝑁𝑉𝑁𝑉)
3 prelpwi 5452 . . . . 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 4790 . . . . . . 7 (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥𝑁)
76adantl 481 . . . . . 6 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥𝑁)
8 eqidd 2738 . . . . . 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 3003 . . . . . . . 8 (𝑎 = 𝑥 → (𝑎𝑁𝑥𝑁))
12 preq1 4733 . . . . . . . . 9 (𝑎 = 𝑥 → {𝑎, 𝑁} = {𝑥, 𝑁})
1312eqeq2d 2748 . . . . . . . 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 2833 . . . . 5 ({𝑥, 𝑁} ∈ 𝑃 ↔ {𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})})
20 eqeq1 2741 . . . . . . . 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 3179 . . . . . 6 (𝑣 = {𝑥, 𝑁} → (∃𝑎𝑉 (𝑎𝑁𝑣 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
23 eqeq1 2741 . . . . . . . . 9 (𝑥 = 𝑣 → (𝑥 = {𝑎, 𝑁} ↔ 𝑣 = {𝑎, 𝑁}))
2423anbi2d 630 . . . . . . . 8 (𝑥 = 𝑣 → ((𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ (𝑎𝑁𝑣 = {𝑎, 𝑁})))
2524rexbidv 3179 . . . . . . 7 (𝑥 = 𝑣 → (∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁𝑣 = {𝑎, 𝑁})))
2625cbvrabv 3447 . . . . . 6 {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})} = {𝑣 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑣 = {𝑎, 𝑁})}
2722, 26elrab2 3695 . . . . 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 3146 . 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 4786 . . . . . . . . 9 (𝑎 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑎𝑉𝑎𝑁))
3533, 34sylibr 234 . . . . . . . 8 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → 𝑎 ∈ (𝑉 ∖ {𝑁}))
36 eqeq1 2741 . . . . . . . . . . . . . 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 3484 . . . . . . . . . . . . . 14 𝑎 ∈ V
40 vex 3484 . . . . . . . . . . . . . 14 𝑥 ∈ V
4139, 40preqr1 4848 . . . . . . . . . . . . 13 ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑎 = 𝑥)
4241equcomd 2018 . . . . . . . . . . . 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 2019 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → {𝑎, 𝑁} = {𝑥, 𝑁})
4645eqeq2d 2748 . . . . . . . . . . . . . 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 3146 . . . . . . . 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 3164 . . . . 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 2741 . . . . . . 7 (𝑥 = 𝑒 → (𝑥 = {𝑎, 𝑁} ↔ 𝑒 = {𝑎, 𝑁}))
5857anbi2d 630 . . . . . 6 (𝑥 = 𝑒 → ((𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ (𝑎𝑁𝑒 = {𝑎, 𝑁})))
5958rexbidv 3179 . . . . 5 (𝑥 = 𝑒 → (∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})))
6059, 18elrab2 3695 . . . 4 (𝑒𝑃 ↔ (𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})))
61 reu6 3732 . . . 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 3145 . 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 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  ∃!wreu 3378  {crab 3436  cdif 3948  𝒫 cpw 4600  {csn 4626  {cpr 4628  cmpt 5225  1-1-ontowf1o 6560  cfv 6561  Vtxcvtx 29013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568
This theorem is referenced by:  cusgrfilem3  29475
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