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Theorem cusgrfilem2 27230
Description: Lemma 2 for cusgrfi 27232. (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 4101 . . . . 5 (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥𝑉)
2 id 22 . . . . 5 (𝑁𝑉𝑁𝑉)
3 prelpwi 5330 . . . . 5 ((𝑥𝑉𝑁𝑉) → {𝑥, 𝑁} ∈ 𝒫 𝑉)
41, 2, 3syl2anr 598 . . . 4 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} ∈ 𝒫 𝑉)
51adantl 484 . . . . 5 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥𝑉)
6 eldifsni 4714 . . . . . . 7 (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥𝑁)
76adantl 484 . . . . . 6 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥𝑁)
8 eqidd 2820 . . . . . 6 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} = {𝑥, 𝑁})
97, 8jca 514 . . . . 5 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁}))
10 id 22 . . . . . 6 (𝑥𝑉𝑥𝑉)
11 neeq1 3076 . . . . . . . 8 (𝑎 = 𝑥 → (𝑎𝑁𝑥𝑁))
12 preq1 4661 . . . . . . . . 9 (𝑎 = 𝑥 → {𝑎, 𝑁} = {𝑥, 𝑁})
1312eqeq2d 2830 . . . . . . . 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 484 . . . . . 6 ((𝑥𝑉𝑎 = 𝑥) → ((𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁})))
1610, 15rspcedv 3614 . . . . 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 2902 . . . . 5 ({𝑥, 𝑁} ∈ 𝑃 ↔ {𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})})
20 eqeq1 2823 . . . . . . . 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 3295 . . . . . 6 (𝑣 = {𝑥, 𝑁} → (∃𝑎𝑉 (𝑎𝑁𝑣 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
23 eqeq1 2823 . . . . . . . . 9 (𝑥 = 𝑣 → (𝑥 = {𝑎, 𝑁} ↔ 𝑣 = {𝑎, 𝑁}))
2423anbi2d 630 . . . . . . . 8 (𝑥 = 𝑣 → ((𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ (𝑎𝑁𝑣 = {𝑎, 𝑁})))
2524rexbidv 3295 . . . . . . 7 (𝑥 = 𝑣 → (∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁𝑣 = {𝑎, 𝑁})))
2625cbvrabv 3490 . . . . . 6 {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})} = {𝑣 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑣 = {𝑎, 𝑁})}
2722, 26elrab2 3681 . . . . 5 ({𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})} ↔ ({𝑥, 𝑁} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
2819, 27bitri 277 . . . 4 ({𝑥, 𝑁} ∈ 𝑃 ↔ ({𝑥, 𝑁} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
294, 17, 28sylanbrc 585 . . 3 ((𝑁𝑉𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} ∈ 𝑃)
3029ralrimiva 3180 . 2 (𝑁𝑉 → ∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃)
31 simpl 485 . . . . . . . . . . 11 ((𝑎𝑁𝑒 = {𝑎, 𝑁}) → 𝑎𝑁)
3231anim2i 618 . . . . . . . . . 10 ((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) → (𝑎𝑉𝑎𝑁))
3332adantl 484 . . . . . . . . 9 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → (𝑎𝑉𝑎𝑁))
34 eldifsn 4711 . . . . . . . . 9 (𝑎 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑎𝑉𝑎𝑁))
3533, 34sylibr 236 . . . . . . . 8 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → 𝑎 ∈ (𝑉 ∖ {𝑁}))
36 eqeq1 2823 . . . . . . . . . . . . . 14 (𝑒 = {𝑎, 𝑁} → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
3736adantl 484 . . . . . . . . . . . . 13 ((𝑎𝑁𝑒 = {𝑎, 𝑁}) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
3837ad2antlr 725 . . . . . . . . . . . 12 (((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
39 vex 3496 . . . . . . . . . . . . . 14 𝑎 ∈ V
40 vex 3496 . . . . . . . . . . . . . 14 𝑥 ∈ V
4139, 40preqr1 4771 . . . . . . . . . . . . 13 ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑎 = 𝑥)
4241equcomd 2019 . . . . . . . . . . . 12 ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑥 = 𝑎)
4338, 42syl6bi 255 . . . . . . . . . . 11 (((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
4443adantll 712 . . . . . . . . . 10 ((((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
4512equcoms 2020 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → {𝑎, 𝑁} = {𝑥, 𝑁})
4645eqeq2d 2830 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑒 = {𝑎, 𝑁} ↔ 𝑒 = {𝑥, 𝑁}))
4746biimpcd 251 . . . . . . . . . . . . 13 (𝑒 = {𝑎, 𝑁} → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
4847adantl 484 . . . . . . . . . . . 12 ((𝑎𝑁𝑒 = {𝑎, 𝑁}) → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
4948adantl 484 . . . . . . . . . . 11 ((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
5049ad2antlr 725 . . . . . . . . . 10 ((((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
5144, 50impbid 214 . . . . . . . . 9 ((((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
5251ralrimiva 3180 . . . . . . . 8 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
5335, 52jca 514 . . . . . . 7 (((𝑁𝑉𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
5453ex 415 . . . . . 6 ((𝑁𝑉𝑒 ∈ 𝒫 𝑉) → ((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))))
5554reximdv2 3269 . . . . 5 ((𝑁𝑉𝑒 ∈ 𝒫 𝑉) → (∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁}) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
5655expimpd 456 . . . 4 (𝑁𝑉 → ((𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
57 eqeq1 2823 . . . . . . 7 (𝑥 = 𝑒 → (𝑥 = {𝑎, 𝑁} ↔ 𝑒 = {𝑎, 𝑁}))
5857anbi2d 630 . . . . . 6 (𝑥 = 𝑒 → ((𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ (𝑎𝑁𝑒 = {𝑎, 𝑁})))
5958rexbidv 3295 . . . . 5 (𝑥 = 𝑒 → (∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})))
6059, 18elrab2 3681 . . . 4 (𝑒𝑃 ↔ (𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})))
61 reu6 3715 . . . 4 (∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁} ↔ ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
6256, 60, 613imtr4g 298 . . 3 (𝑁𝑉 → (𝑒𝑃 → ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
6362ralrimiv 3179 . 2 (𝑁𝑉 → ∀𝑒𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁})
64 cusgrfi.f . . 3 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})
6564f1ompt 6868 . 2 (𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃 ↔ (∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃 ∧ ∀𝑒𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
6630, 63, 65sylanbrc 585 1 (𝑁𝑉𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  wne 3014  wral 3136  wrex 3137  ∃!wreu 3138  {crab 3140  cdif 3931  𝒫 cpw 4537  {csn 4559  {cpr 4561  cmpt 5137  1-1-ontowf1o 6347  cfv 6348  Vtxcvtx 26773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  cusgrfilem3  27231
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