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

Step	Hyp	Ref	Expression
1		ax-1ne0 9884	. . 3 ⊢ 1 ≠ 0
2	1	neii 2784	. 2 ⊢ ¬ 1 = 0
3		0ne1 10965	. . . . 5 ⊢ 0 ≠ 1
4		c0ex 9913	. . . . . 6 ⊢ 0 ∈ V
5		1ex 9914	. . . . . 6 ⊢ 1 ∈ V
6	4, 5, 4, 4	fpr 6326	. . . . 5 ⊢ (0 ≠ 1 → {⟨0, 0⟩, ⟨1, 0⟩}:{0, 1}⟶{0, 0})
7	3, 6	ax-mp 5	. . . 4 ⊢ {⟨0, 0⟩, ⟨1, 0⟩}:{0, 1}⟶{0, 0}
8		wlkntrl.f	. . . . . 6 ⊢ 𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}
9	8	eqcomi 2619	. . . . 5 ⊢ {⟨0, 0⟩, ⟨1, 0⟩} = 𝐹
10	9	feq1i 5949	. . . 4 ⊢ ({⟨0, 0⟩, ⟨1, 0⟩}:{0, 1}⟶{0, 0} ↔ 𝐹:{0, 1}⟶{0, 0})
11	7, 10	mpbi 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} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
14		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘0))
15	14	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘0) = (𝐹‘𝑦)))
16		eqeq1 2614	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
17	15, 16	imbi12d 333	. . . . . . . . 9 ⊢ (𝑥 = 0 → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘0) = (𝐹‘𝑦) → 0 = 𝑦)))
18	17	ralbidv 2969	. . . . . . . 8 ⊢ (𝑥 = 0 → (∀𝑦 ∈ {0, 1} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {0, 1} ((𝐹‘0) = (𝐹‘𝑦) → 0 = 𝑦)))
19		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝐹‘𝑥) = (𝐹‘1))
20	19	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘1) = (𝐹‘𝑦)))
21		eqeq1 2614	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑥 = 𝑦 ↔ 1 = 𝑦))
22	20, 21	imbi12d 333	. . . . . . . . 9 ⊢ (𝑥 = 1 → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘1) = (𝐹‘𝑦) → 1 = 𝑦)))
23	22	ralbidv 2969	. . . . . . . 8 ⊢ (𝑥 = 1 → (∀𝑦 ∈ {0, 1} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {0, 1} ((𝐹‘1) = (𝐹‘𝑦) → 1 = 𝑦)))
24	4, 5, 18, 23	ralpr 4185	. . . . . . 7 ⊢ (∀𝑥 ∈ {0, 1}∀𝑦 ∈ {0, 1} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (∀𝑦 ∈ {0, 1} ((𝐹‘0) = (𝐹‘𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1} ((𝐹‘1) = (𝐹‘𝑦) → 1 = 𝑦)))
25		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (𝐹‘𝑦) = (𝐹‘0))
26	25	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑦 = 0 → ((𝐹‘0) = (𝐹‘𝑦) ↔ (𝐹‘0) = (𝐹‘0)))
27		eqeq2 2621	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (0 = 𝑦 ↔ 0 = 0))
28	26, 27	imbi12d 333	. . . . . . . . 9 ⊢ (𝑦 = 0 → (((𝐹‘0) = (𝐹‘𝑦) → 0 = 𝑦) ↔ ((𝐹‘0) = (𝐹‘0) → 0 = 0)))
29		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → (𝐹‘𝑦) = (𝐹‘1))
30	29	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑦 = 1 → ((𝐹‘0) = (𝐹‘𝑦) ↔ (𝐹‘0) = (𝐹‘1)))
31		eqeq2 2621	. . . . . . . . . 10 ⊢ (𝑦 = 1 → (0 = 𝑦 ↔ 0 = 1))
32	30, 31	imbi12d 333	. . . . . . . . 9 ⊢ (𝑦 = 1 → (((𝐹‘0) = (𝐹‘𝑦) → 0 = 𝑦) ↔ ((𝐹‘0) = (𝐹‘1) → 0 = 1)))
33	4, 5, 28, 32	ralpr 4185	. . . . . . . 8 ⊢ (∀𝑦 ∈ {0, 1} ((𝐹‘0) = (𝐹‘𝑦) → 0 = 𝑦) ↔ (((𝐹‘0) = (𝐹‘0) → 0 = 0) ∧ ((𝐹‘0) = (𝐹‘1) → 0 = 1)))
34	25	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑦 = 0 → ((𝐹‘1) = (𝐹‘𝑦) ↔ (𝐹‘1) = (𝐹‘0)))
35		eqeq2 2621	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (1 = 𝑦 ↔ 1 = 0))
36	34, 35	imbi12d 333	. . . . . . . . 9 ⊢ (𝑦 = 0 → (((𝐹‘1) = (𝐹‘𝑦) → 1 = 𝑦) ↔ ((𝐹‘1) = (𝐹‘0) → 1 = 0)))
37	29	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑦 = 1 → ((𝐹‘1) = (𝐹‘𝑦) ↔ (𝐹‘1) = (𝐹‘1)))
38		eqeq2 2621	. . . . . . . . . 10 ⊢ (𝑦 = 1 → (1 = 𝑦 ↔ 1 = 1))
39	37, 38	imbi12d 333	. . . . . . . . 9 ⊢ (𝑦 = 1 → (((𝐹‘1) = (𝐹‘𝑦) → 1 = 𝑦) ↔ ((𝐹‘1) = (𝐹‘1) → 1 = 1)))
40	4, 5, 36, 39	ralpr 4185	. . . . . . . 8 ⊢ (∀𝑦 ∈ {0, 1} ((𝐹‘1) = (𝐹‘𝑦) → 1 = 𝑦) ↔ (((𝐹‘1) = (𝐹‘0) → 1 = 0) ∧ ((𝐹‘1) = (𝐹‘1) → 1 = 1)))
41	8	fveq1i 6104	. . . . . . . . . . . . 13 ⊢ (𝐹‘1) = ({⟨0, 0⟩, ⟨1, 0⟩}‘1)
42	5, 4	fvpr2 6362	. . . . . . . . . . . . . 14 ⊢ (0 ≠ 1 → ({⟨0, 0⟩, ⟨1, 0⟩}‘1) = 0)
43	3, 42	mp1i 13	. . . . . . . . . . . . 13 ⊢ (¬ 1 = 0 → ({⟨0, 0⟩, ⟨1, 0⟩}‘1) = 0)
44	41, 43	syl5eq 2656	. . . . . . . . . . . 12 ⊢ (¬ 1 = 0 → (𝐹‘1) = 0)
45	8	fveq1i 6104	. . . . . . . . . . . . 13 ⊢ (𝐹‘0) = ({⟨0, 0⟩, ⟨1, 0⟩}‘0)
46	4, 4	fvpr1 6361	. . . . . . . . . . . . . 14 ⊢ (0 ≠ 1 → ({⟨0, 0⟩, ⟨1, 0⟩}‘0) = 0)
47	3, 46	mp1i 13	. . . . . . . . . . . . 13 ⊢ (¬ 1 = 0 → ({⟨0, 0⟩, ⟨1, 0⟩}‘0) = 0)
48	45, 47	syl5req 2657	. . . . . . . . . . . 12 ⊢ (¬ 1 = 0 → 0 = (𝐹‘0))
49	44, 48	eqtrd 2644	. . . . . . . . . . 11 ⊢ (¬ 1 = 0 → (𝐹‘1) = (𝐹‘0))
50	49	con1i 143	. . . . . . . . . 10 ⊢ (¬ (𝐹‘1) = (𝐹‘0) → 1 = 0)
51		id 22	. . . . . . . . . 10 ⊢ (1 = 0 → 1 = 0)
52	50, 51	ja 172	. . . . . . . . 9 ⊢ (((𝐹‘1) = (𝐹‘0) → 1 = 0) → 1 = 0)
53	52	ad2antrl 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)
54	33, 40, 53	syl2anb 495	. . . . . . 7 ⊢ ((∀𝑦 ∈ {0, 1} ((𝐹‘0) = (𝐹‘𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1} ((𝐹‘1) = (𝐹‘𝑦) → 1 = 𝑦)) → 1 = 0)
55	24, 54	sylbi 206	. . . . . 6 ⊢ (∀𝑥 ∈ {0, 1}∀𝑦 ∈ {0, 1} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) → 1 = 0)
56	55	adantl 481	. . . . 5 ⊢ ((𝐹:{0, 1}⟶{0, 0} ∧ ∀𝑥 ∈ {0, 1}∀𝑦 ∈ {0, 1} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) → 1 = 0)
57	13, 56	sylbi 206	. . . 4 ⊢ (𝐹:{0, 1}–1-1→{0, 0} → 1 = 0)
58	12, 57	sylbir 224	. . 3 ⊢ ((𝐹:{0, 1}⟶{0, 0} ∧ Fun ◡𝐹) → 1 = 0)
59	11, 58	mpan 702	. 2 ⊢ (Fun ◡𝐹 → 1 = 0)
60	2, 59	mto 187	1 ⊢ ¬ Fun ◡𝐹

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