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Theorem wlkntrllem3 26091
 Description: Lemma 3 for wlkntrl 26092: F is not injective. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
Hypotheses
Ref Expression
wlkntrl.v 𝑉 = {𝑥, 𝑦}
wlkntrl.e 𝐸 = {⟨0, {𝑥, 𝑦}⟩}
wlkntrl.f 𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}
wlkntrl.p 𝑃 = {⟨0, 𝑥⟩, ⟨1, 𝑦⟩, ⟨2, 𝑥⟩}
Assertion
Ref Expression
wlkntrllem3 ¬ Fun 𝐹
Distinct variable group:   𝑥,𝐹,𝑦
Allowed substitution hints:   𝑃(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem wlkntrllem3
StepHypRef Expression
1 ax-1ne0 9884 . . 3 1 ≠ 0
21neii 2784 . 2 ¬ 1 = 0
3 0ne1 10965 . . . . 5 0 ≠ 1
4 c0ex 9913 . . . . . 6 0 ∈ V
5 1ex 9914 . . . . . 6 1 ∈ V
64, 5, 4, 4fpr 6326 . . . . 5 (0 ≠ 1 → {⟨0, 0⟩, ⟨1, 0⟩}:{0, 1}⟶{0, 0})
73, 6ax-mp 5 . . . 4 {⟨0, 0⟩, ⟨1, 0⟩}:{0, 1}⟶{0, 0}
8 wlkntrl.f . . . . . 6 𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}
98eqcomi 2619 . . . . 5 {⟨0, 0⟩, ⟨1, 0⟩} = 𝐹
109feq1i 5949 . . . 4 ({⟨0, 0⟩, ⟨1, 0⟩}:{0, 1}⟶{0, 0} ↔ 𝐹:{0, 1}⟶{0, 0})
117, 10mpbi 219 . . 3 𝐹:{0, 1}⟶{0, 0}
12 df-f1 5809 . . . 4 (𝐹:{0, 1}–1-1→{0, 0} ↔ (𝐹:{0, 1}⟶{0, 0} ∧ Fun 𝐹))
13 dff13 6416 . . . . 5 (𝐹:{0, 1}–1-1→{0, 0} ↔ (𝐹:{0, 1}⟶{0, 0} ∧ ∀𝑥 ∈ {0, 1}∀𝑦 ∈ {0, 1} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
14 fveq2 6103 . . . . . . . . . . 11 (𝑥 = 0 → (𝐹𝑥) = (𝐹‘0))
1514eqeq1d 2612 . . . . . . . . . 10 (𝑥 = 0 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹‘0) = (𝐹𝑦)))
16 eqeq1 2614 . . . . . . . . . 10 (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
1715, 16imbi12d 333 . . . . . . . . 9 (𝑥 = 0 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦)))
1817ralbidv 2969 . . . . . . . 8 (𝑥 = 0 → (∀𝑦 ∈ {0, 1} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {0, 1} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦)))
19 fveq2 6103 . . . . . . . . . . 11 (𝑥 = 1 → (𝐹𝑥) = (𝐹‘1))
2019eqeq1d 2612 . . . . . . . . . 10 (𝑥 = 1 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹‘1) = (𝐹𝑦)))
21 eqeq1 2614 . . . . . . . . . 10 (𝑥 = 1 → (𝑥 = 𝑦 ↔ 1 = 𝑦))
2220, 21imbi12d 333 . . . . . . . . 9 (𝑥 = 1 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦)))
2322ralbidv 2969 . . . . . . . 8 (𝑥 = 1 → (∀𝑦 ∈ {0, 1} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {0, 1} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦)))
244, 5, 18, 23ralpr 4185 . . . . . . 7 (∀𝑥 ∈ {0, 1}∀𝑦 ∈ {0, 1} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (∀𝑦 ∈ {0, 1} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦)))
25 fveq2 6103 . . . . . . . . . . 11 (𝑦 = 0 → (𝐹𝑦) = (𝐹‘0))
2625eqeq2d 2620 . . . . . . . . . 10 (𝑦 = 0 → ((𝐹‘0) = (𝐹𝑦) ↔ (𝐹‘0) = (𝐹‘0)))
27 eqeq2 2621 . . . . . . . . . 10 (𝑦 = 0 → (0 = 𝑦 ↔ 0 = 0))
2826, 27imbi12d 333 . . . . . . . . 9 (𝑦 = 0 → (((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ ((𝐹‘0) = (𝐹‘0) → 0 = 0)))
29 fveq2 6103 . . . . . . . . . . 11 (𝑦 = 1 → (𝐹𝑦) = (𝐹‘1))
3029eqeq2d 2620 . . . . . . . . . 10 (𝑦 = 1 → ((𝐹‘0) = (𝐹𝑦) ↔ (𝐹‘0) = (𝐹‘1)))
31 eqeq2 2621 . . . . . . . . . 10 (𝑦 = 1 → (0 = 𝑦 ↔ 0 = 1))
3230, 31imbi12d 333 . . . . . . . . 9 (𝑦 = 1 → (((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ ((𝐹‘0) = (𝐹‘1) → 0 = 1)))
334, 5, 28, 32ralpr 4185 . . . . . . . 8 (∀𝑦 ∈ {0, 1} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ (((𝐹‘0) = (𝐹‘0) → 0 = 0) ∧ ((𝐹‘0) = (𝐹‘1) → 0 = 1)))
3425eqeq2d 2620 . . . . . . . . . 10 (𝑦 = 0 → ((𝐹‘1) = (𝐹𝑦) ↔ (𝐹‘1) = (𝐹‘0)))
35 eqeq2 2621 . . . . . . . . . 10 (𝑦 = 0 → (1 = 𝑦 ↔ 1 = 0))
3634, 35imbi12d 333 . . . . . . . . 9 (𝑦 = 0 → (((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ ((𝐹‘1) = (𝐹‘0) → 1 = 0)))
3729eqeq2d 2620 . . . . . . . . . 10 (𝑦 = 1 → ((𝐹‘1) = (𝐹𝑦) ↔ (𝐹‘1) = (𝐹‘1)))
38 eqeq2 2621 . . . . . . . . . 10 (𝑦 = 1 → (1 = 𝑦 ↔ 1 = 1))
3937, 38imbi12d 333 . . . . . . . . 9 (𝑦 = 1 → (((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ ((𝐹‘1) = (𝐹‘1) → 1 = 1)))
404, 5, 36, 39ralpr 4185 . . . . . . . 8 (∀𝑦 ∈ {0, 1} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ (((𝐹‘1) = (𝐹‘0) → 1 = 0) ∧ ((𝐹‘1) = (𝐹‘1) → 1 = 1)))
418fveq1i 6104 . . . . . . . . . . . . 13 (𝐹‘1) = ({⟨0, 0⟩, ⟨1, 0⟩}‘1)
425, 4fvpr2 6362 . . . . . . . . . . . . . 14 (0 ≠ 1 → ({⟨0, 0⟩, ⟨1, 0⟩}‘1) = 0)
433, 42mp1i 13 . . . . . . . . . . . . 13 (¬ 1 = 0 → ({⟨0, 0⟩, ⟨1, 0⟩}‘1) = 0)
4441, 43syl5eq 2656 . . . . . . . . . . . 12 (¬ 1 = 0 → (𝐹‘1) = 0)
458fveq1i 6104 . . . . . . . . . . . . 13 (𝐹‘0) = ({⟨0, 0⟩, ⟨1, 0⟩}‘0)
464, 4fvpr1 6361 . . . . . . . . . . . . . 14 (0 ≠ 1 → ({⟨0, 0⟩, ⟨1, 0⟩}‘0) = 0)
473, 46mp1i 13 . . . . . . . . . . . . 13 (¬ 1 = 0 → ({⟨0, 0⟩, ⟨1, 0⟩}‘0) = 0)
4845, 47syl5req 2657 . . . . . . . . . . . 12 (¬ 1 = 0 → 0 = (𝐹‘0))
4944, 48eqtrd 2644 . . . . . . . . . . 11 (¬ 1 = 0 → (𝐹‘1) = (𝐹‘0))
5049con1i 143 . . . . . . . . . 10 (¬ (𝐹‘1) = (𝐹‘0) → 1 = 0)
51 id 22 . . . . . . . . . 10 (1 = 0 → 1 = 0)
5250, 51ja 172 . . . . . . . . 9 (((𝐹‘1) = (𝐹‘0) → 1 = 0) → 1 = 0)
5352ad2antrl 760 . . . . . . . 8 (((((𝐹‘0) = (𝐹‘0) → 0 = 0) ∧ ((𝐹‘0) = (𝐹‘1) → 0 = 1)) ∧ (((𝐹‘1) = (𝐹‘0) → 1 = 0) ∧ ((𝐹‘1) = (𝐹‘1) → 1 = 1))) → 1 = 0)
5433, 40, 53syl2anb 495 . . . . . . 7 ((∀𝑦 ∈ {0, 1} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦)) → 1 = 0)
5524, 54sylbi 206 . . . . . 6 (∀𝑥 ∈ {0, 1}∀𝑦 ∈ {0, 1} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) → 1 = 0)
5655adantl 481 . . . . 5 ((𝐹:{0, 1}⟶{0, 0} ∧ ∀𝑥 ∈ {0, 1}∀𝑦 ∈ {0, 1} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) → 1 = 0)
5713, 56sylbi 206 . . . 4 (𝐹:{0, 1}–1-1→{0, 0} → 1 = 0)
5812, 57sylbir 224 . . 3 ((𝐹:{0, 1}⟶{0, 0} ∧ Fun 𝐹) → 1 = 0)
5911, 58mpan 702 . 2 (Fun 𝐹 → 1 = 0)
602, 59mto 187 1 ¬ Fun 𝐹
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ≠ wne 2780  ∀wral 2896  {csn 4125  {cpr 4127  {ctp 4129  ⟨cop 4131  ◡ccnv 5037  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  0cc0 9815  1c1 9816  2c2 10947 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-8 1979  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-pow 4769  ax-pr 4833  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-mulcl 9877  ax-i2m1 9883  ax-1ne0 9884 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-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-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812 This theorem is referenced by:  wlkntrl  26092
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