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Theorem cusgrfilem2 26273
Description: Lemma 2 for cusgrfi 26275. (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 3716 . . . . 5 (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥𝑉)
2 id 22 . . . . 5 (𝑁𝑉𝑁𝑉)
3 prelpwi 4886 . . . . 5 ((𝑥𝑉𝑁𝑉) → {𝑥, 𝑁} ∈ 𝒫 𝑉)
41, 2, 3syl2anr 495 . . . 4 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} ∈ 𝒫 𝑉)
51adantl 482 . . . . 5 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥𝑉)
6 eldifsni 4296 . . . . . . 7 (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥𝑁)
76adantl 482 . . . . . 6 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥𝑁)
8 eqidd 2622 . . . . . 6 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} = {𝑥, 𝑁})
97, 8jca 554 . . . . 5 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁}))
10 id 22 . . . . . 6 (𝑥𝑉𝑥𝑉)
11 neeq1 2852 . . . . . . . 8 (𝑎 = 𝑥 → (𝑎𝑁𝑥𝑁))
12 preq1 4245 . . . . . . . . 9 (𝑎 = 𝑥 → {𝑎, 𝑁} = {𝑥, 𝑁})
1312eqeq2d 2631 . . . . . . . 8 (𝑎 = 𝑥 → ({𝑥, 𝑁} = {𝑎, 𝑁} ↔ {𝑥, 𝑁} = {𝑥, 𝑁}))
1411, 13anbi12d 746 . . . . . . 7 (𝑎 = 𝑥 → ((𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁})))
1514adantl 482 . . . . . 6 ((𝑥𝑉𝑎 = 𝑥) → ((𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁})))
1610, 15rspcedv 3303 . . . . 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 2690 . . . . 5 ({𝑥, 𝑁} ∈ 𝑃 ↔ {𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})})
20 eqeq1 2625 . . . . . . . 8 (𝑣 = {𝑥, 𝑁} → (𝑣 = {𝑎, 𝑁} ↔ {𝑥, 𝑁} = {𝑎, 𝑁}))
2120anbi2d 739 . . . . . . 7 (𝑣 = {𝑥, 𝑁} → ((𝑎𝑁𝑣 = {𝑎, 𝑁}) ↔ (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
2221rexbidv 3047 . . . . . 6 (𝑣 = {𝑥, 𝑁} → (∃𝑎𝑉 (𝑎𝑁𝑣 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
23 eqeq1 2625 . . . . . . . . 9 (𝑥 = 𝑣 → (𝑥 = {𝑎, 𝑁} ↔ 𝑣 = {𝑎, 𝑁}))
2423anbi2d 739 . . . . . . . 8 (𝑥 = 𝑣 → ((𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ (𝑎𝑁𝑣 = {𝑎, 𝑁})))
2524rexbidv 3047 . . . . . . 7 (𝑥 = 𝑣 → (∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁𝑣 = {𝑎, 𝑁})))
2625cbvrabv 3189 . . . . . 6 {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})} = {𝑣 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑣 = {𝑎, 𝑁})}
2722, 26elrab2 3353 . . . . 5 ({𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})} ↔ ({𝑥, 𝑁} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
2819, 27bitri 264 . . . 4 ({𝑥, 𝑁} ∈ 𝑃 ↔ ({𝑥, 𝑁} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
294, 17, 28sylanbrc 697 . . 3 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} ∈ 𝑃)
3029ralrimiva 2962 . 2 (𝑁𝑉 → ∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃)
31 simpl 473 . . . . . . . . . . 11 ((𝑎𝑁𝑒 = {𝑎, 𝑁}) → 𝑎𝑁)
3231anim2i 592 . . . . . . . . . 10 ((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) → (𝑎𝑉𝑎𝑁))
3332adantl 482 . . . . . . . . 9 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → (𝑎𝑉𝑎𝑁))
34 eldifsn 4294 . . . . . . . . 9 (𝑎 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑎𝑉𝑎𝑁))
3533, 34sylibr 224 . . . . . . . 8 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → 𝑎 ∈ (𝑉 ∖ {𝑁}))
36 eqeq1 2625 . . . . . . . . . . . . . 14 (𝑒 = {𝑎, 𝑁} → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
3736adantl 482 . . . . . . . . . . . . 13 ((𝑎𝑁𝑒 = {𝑎, 𝑁}) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
3837ad2antlr 762 . . . . . . . . . . . 12 (((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
39 vex 3193 . . . . . . . . . . . . . 14 𝑎 ∈ V
40 vex 3193 . . . . . . . . . . . . . 14 𝑥 ∈ V
4139, 40preqr1 4354 . . . . . . . . . . . . 13 ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑎 = 𝑥)
4241equcomd 1943 . . . . . . . . . . . 12 ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑥 = 𝑎)
4338, 42syl6bi 243 . . . . . . . . . . 11 (((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
4443adantll 749 . . . . . . . . . 10 ((((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
4512equcoms 1944 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → {𝑎, 𝑁} = {𝑥, 𝑁})
4645eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑒 = {𝑎, 𝑁} ↔ 𝑒 = {𝑥, 𝑁}))
4746biimpcd 239 . . . . . . . . . . . . 13 (𝑒 = {𝑎, 𝑁} → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
4847adantl 482 . . . . . . . . . . . 12 ((𝑎𝑁𝑒 = {𝑎, 𝑁}) → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
4948adantl 482 . . . . . . . . . . 11 ((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
5049ad2antlr 762 . . . . . . . . . 10 ((((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
5144, 50impbid 202 . . . . . . . . 9 ((((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
5251ralrimiva 2962 . . . . . . . 8 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
5335, 52jca 554 . . . . . . 7 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
5453ex 450 . . . . . 6 ((𝑁𝑉𝑒 ∈ 𝒫 𝑉) → ((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))))
5554reximdv2 3010 . . . . 5 ((𝑁𝑉𝑒 ∈ 𝒫 𝑉) → (∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁}) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
5655expimpd 628 . . . 4 (𝑁𝑉 → ((𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
57 eqeq1 2625 . . . . . . 7 (𝑥 = 𝑒 → (𝑥 = {𝑎, 𝑁} ↔ 𝑒 = {𝑎, 𝑁}))
5857anbi2d 739 . . . . . 6 (𝑥 = 𝑒 → ((𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ (𝑎𝑁𝑒 = {𝑎, 𝑁})))
5958rexbidv 3047 . . . . 5 (𝑥 = 𝑒 → (∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})))
6059, 18elrab2 3353 . . . 4 (𝑒𝑃 ↔ (𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})))
61 reu6 3382 . . . 4 (∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁} ↔ ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
6256, 60, 613imtr4g 285 . . 3 (𝑁𝑉 → (𝑒𝑃 → ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
6362ralrimiv 2961 . 2 (𝑁𝑉 → ∀𝑒𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁})
64 cusgrfi.f . . 3 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})
6564f1ompt 6348 . 2 (𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃 ↔ (∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃 ∧ ∀𝑒𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
6630, 63, 65sylanbrc 697 1 (𝑁𝑉𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  ∃!wreu 2910  {crab 2912  cdif 3557  𝒫 cpw 4136  {csn 4155  {cpr 4157  cmpt 4683  1-1-ontowf1o 5856  cfv 5857  Vtxcvtx 25808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865
This theorem is referenced by:  cusgrfilem3  26274
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