Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cusgrafilem2 Structured version   Visualization version   GIF version

Theorem cusgrafilem2 26008
 Description: Lemma 2 for cusgrafi 26010. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
Hypotheses
Ref Expression
cusgrafi.p 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}
cusgrafi.f 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})
Assertion
Ref Expression
cusgrafilem2 ((𝑉𝑊𝑁𝑉) → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃)
Distinct variable groups:   𝑁,𝑎,𝑥   𝑉,𝑎,𝑥   𝑥,𝑃   𝑊,𝑎,𝑥
Allowed substitution hints:   𝑃(𝑎)   𝐹(𝑥,𝑎)

Proof of Theorem cusgrafilem2
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 4260 . . . . . . 7 (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣𝑉𝑣𝑁))
2 simpl 472 . . . . . . 7 ((𝑣𝑉𝑣𝑁) → 𝑣𝑉)
31, 2sylbi 206 . . . . . 6 (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣𝑉)
4 simpr 476 . . . . . 6 ((𝑉𝑊𝑁𝑉) → 𝑁𝑉)
5 prelpwi 4842 . . . . . 6 ((𝑣𝑉𝑁𝑉) → {𝑣, 𝑁} ∈ 𝒫 𝑉)
63, 4, 5syl2anr 494 . . . . 5 (((𝑉𝑊𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → {𝑣, 𝑁} ∈ 𝒫 𝑉)
71biimpi 205 . . . . . . 7 (𝑣 ∈ (𝑉 ∖ {𝑁}) → (𝑣𝑉𝑣𝑁))
87adantl 481 . . . . . 6 (((𝑉𝑊𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (𝑣𝑉𝑣𝑁))
9 simpr 476 . . . . . . . . 9 ((𝑣𝑉𝑣𝑁) → 𝑣𝑁)
101, 9sylbi 206 . . . . . . . 8 (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣𝑁)
1110adantl 481 . . . . . . 7 (((𝑉𝑊𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣𝑁)
12 eqidd 2611 . . . . . . 7 (((𝑉𝑊𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → {𝑣, 𝑁} = {𝑣, 𝑁})
1311, 12jca 553 . . . . . 6 (((𝑉𝑊𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (𝑣𝑁 ∧ {𝑣, 𝑁} = {𝑣, 𝑁}))
14 neeq1 2844 . . . . . . . . 9 (𝑎 = 𝑣 → (𝑎𝑁𝑣𝑁))
15 preq1 4212 . . . . . . . . . 10 (𝑎 = 𝑣 → {𝑎, 𝑁} = {𝑣, 𝑁})
1615eqeq2d 2620 . . . . . . . . 9 (𝑎 = 𝑣 → ({𝑣, 𝑁} = {𝑎, 𝑁} ↔ {𝑣, 𝑁} = {𝑣, 𝑁}))
1714, 16anbi12d 743 . . . . . . . 8 (𝑎 = 𝑣 → ((𝑎𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁}) ↔ (𝑣𝑁 ∧ {𝑣, 𝑁} = {𝑣, 𝑁})))
1817adantl 481 . . . . . . 7 (((𝑣𝑉𝑣𝑁) ∧ 𝑎 = 𝑣) → ((𝑎𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁}) ↔ (𝑣𝑁 ∧ {𝑣, 𝑁} = {𝑣, 𝑁})))
192, 18rspcedv 3286 . . . . . 6 ((𝑣𝑉𝑣𝑁) → ((𝑣𝑁 ∧ {𝑣, 𝑁} = {𝑣, 𝑁}) → ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁})))
208, 13, 19sylc 63 . . . . 5 (((𝑉𝑊𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁}))
21 eqeq1 2614 . . . . . . . 8 (𝑥 = {𝑣, 𝑁} → (𝑥 = {𝑎, 𝑁} ↔ {𝑣, 𝑁} = {𝑎, 𝑁}))
2221anbi2d 736 . . . . . . 7 (𝑥 = {𝑣, 𝑁} → ((𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ (𝑎𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁})))
2322rexbidv 3034 . . . . . 6 (𝑥 = {𝑣, 𝑁} → (∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁})))
24 cusgrafi.p . . . . . 6 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}
2523, 24elrab2 3333 . . . . 5 ({𝑣, 𝑁} ∈ 𝑃 ↔ ({𝑣, 𝑁} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁})))
266, 20, 25sylanbrc 695 . . . 4 (((𝑉𝑊𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → {𝑣, 𝑁} ∈ 𝑃)
2726ralrimiva 2949 . . 3 ((𝑉𝑊𝑁𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑣, 𝑁} ∈ 𝑃)
28 preq1 4212 . . . . 5 (𝑥 = 𝑣 → {𝑥, 𝑁} = {𝑣, 𝑁})
2928eleq1d 2672 . . . 4 (𝑥 = 𝑣 → ({𝑥, 𝑁} ∈ 𝑃 ↔ {𝑣, 𝑁} ∈ 𝑃))
3029cbvralv 3147 . . 3 (∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃 ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑣, 𝑁} ∈ 𝑃)
3127, 30sylibr 223 . 2 ((𝑉𝑊𝑁𝑉) → ∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃)
32 simpl 472 . . . . . . . . . . 11 ((𝑎𝑁𝑒 = {𝑎, 𝑁}) → 𝑎𝑁)
3332anim2i 591 . . . . . . . . . 10 ((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) → (𝑎𝑉𝑎𝑁))
3433adantl 481 . . . . . . . . 9 ((((𝑉𝑊𝑁𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → (𝑎𝑉𝑎𝑁))
35 eldifsn 4260 . . . . . . . . 9 (𝑎 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑎𝑉𝑎𝑁))
3634, 35sylibr 223 . . . . . . . 8 ((((𝑉𝑊𝑁𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → 𝑎 ∈ (𝑉 ∖ {𝑁}))
37 eqeq1 2614 . . . . . . . . . . . . . 14 (𝑒 = {𝑎, 𝑁} → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
3837adantl 481 . . . . . . . . . . . . 13 ((𝑎𝑁𝑒 = {𝑎, 𝑁}) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
3938ad2antlr 759 . . . . . . . . . . . 12 (((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
40 vex 3176 . . . . . . . . . . . . . 14 𝑎 ∈ V
41 vex 3176 . . . . . . . . . . . . . 14 𝑥 ∈ V
4240, 41preqr1 4319 . . . . . . . . . . . . 13 ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑎 = 𝑥)
4342equcomd 1933 . . . . . . . . . . . 12 ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑥 = 𝑎)
4439, 43syl6bi 242 . . . . . . . . . . 11 (((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
4544adantll 746 . . . . . . . . . 10 (((((𝑉𝑊𝑁𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
46 preq1 4212 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → {𝑎, 𝑁} = {𝑥, 𝑁})
4746equcoms 1934 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → {𝑎, 𝑁} = {𝑥, 𝑁})
4847eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑒 = {𝑎, 𝑁} ↔ 𝑒 = {𝑥, 𝑁}))
4948biimpcd 238 . . . . . . . . . . . . 13 (𝑒 = {𝑎, 𝑁} → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
5049adantl 481 . . . . . . . . . . . 12 ((𝑎𝑁𝑒 = {𝑎, 𝑁}) → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
5150adantl 481 . . . . . . . . . . 11 ((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
5251ad2antlr 759 . . . . . . . . . 10 (((((𝑉𝑊𝑁𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥 = 𝑎𝑒 = {𝑥, 𝑁}))
5345, 52impbid 201 . . . . . . . . 9 (((((𝑉𝑊𝑁𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
5453ralrimiva 2949 . . . . . . . 8 ((((𝑉𝑊𝑁𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
5536, 54jca 553 . . . . . . 7 ((((𝑉𝑊𝑁𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁}))) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
5655ex 449 . . . . . 6 (((𝑉𝑊𝑁𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) → ((𝑎𝑉 ∧ (𝑎𝑁𝑒 = {𝑎, 𝑁})) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))))
5756reximdv2 2997 . . . . 5 (((𝑉𝑊𝑁𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) → (∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁}) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
5857expimpd 627 . . . 4 ((𝑉𝑊𝑁𝑉) → ((𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
59 eqeq1 2614 . . . . . . 7 (𝑥 = 𝑒 → (𝑥 = {𝑎, 𝑁} ↔ 𝑒 = {𝑎, 𝑁}))
6059anbi2d 736 . . . . . 6 (𝑥 = 𝑒 → ((𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ (𝑎𝑁𝑒 = {𝑎, 𝑁})))
6160rexbidv 3034 . . . . 5 (𝑥 = 𝑒 → (∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})))
6261, 24elrab2 3333 . . . 4 (𝑒𝑃 ↔ (𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉 (𝑎𝑁𝑒 = {𝑎, 𝑁})))
63 reu6 3362 . . . 4 (∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁} ↔ ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
6458, 62, 633imtr4g 284 . . 3 ((𝑉𝑊𝑁𝑉) → (𝑒𝑃 → ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
6564ralrimiv 2948 . 2 ((𝑉𝑊𝑁𝑉) → ∀𝑒𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁})
66 cusgrafi.f . . 3 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})
6766f1ompt 6290 . 2 (𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃 ↔ (∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃 ∧ ∀𝑒𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
6831, 65, 67sylanbrc 695 1 ((𝑉𝑊𝑁𝑉) → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {crab 2900   ∖ cdif 3537  𝒫 cpw 4108  {csn 4125  {cpr 4127   ↦ cmpt 4643  –1-1-onto→wf1o 5803 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by:  cusgrafilem3  26009
 Copyright terms: Public domain W3C validator