Theorem fnwe2lem3 43837

Description: Lemma for fnwe2 43838. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Hypotheses

Ref	Expression
fnwe2.su	⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)
fnwe2.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}
fnwe2.s	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
fnwe2.f	⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵)
fnwe2.r	⊢ (𝜑 → 𝑅 We 𝐵)
fnwe2lem3.a	⊢ (𝜑 → 𝑎 ∈ 𝐴)
fnwe2lem3.b	⊢ (𝜑 → 𝑏 ∈ 𝐴)

Assertion

Ref	Expression
fnwe2lem3	⊢ (𝜑 → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))

Distinct variable groups:   𝑦,𝑈,𝑧,𝑎,𝑏   𝑥,𝑆,𝑦,𝑎,𝑏   𝑥,𝑅,𝑦,𝑎,𝑏   𝜑,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧,𝑎,𝑏   𝑥,𝐹,𝑦,𝑧,𝑎,𝑏   𝑇,𝑎,𝑏   𝐵,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑎, 𝑏)   𝐵(𝑥, 𝑦, 𝑧)   𝑅(𝑧)   𝑆(𝑧)   𝑇(𝑥, 𝑦, 𝑧)   𝑈(𝑥)

Proof of Theorem fnwe2lem3

Step	Hyp	Ref	Expression
1		animorrl 996	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑎)𝑅(𝐹‘𝑏)) → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))
2		fnwe2.su	. . . . 5 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)
3		fnwe2.t	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}
4	2, 3	fnwe2val 43834	. . . 4 ⊢ (𝑎𝑇𝑏 ↔ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))
5	1, 4	sylibr 237	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑎)𝑅(𝐹‘𝑏)) → 𝑎𝑇𝑏)
6	5	3mix1d 1355	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑎)𝑅(𝐹‘𝑏)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
7		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → (𝐹‘𝑎) = (𝐹‘𝑏))
8		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)
9	7, 8	jca 521	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏))
10	9	olcd 888	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))
11	10, 4	sylibr 237	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → 𝑎𝑇𝑏)
12	11	3mix1d 1355	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
13		3mix2 1350	. . . 4 ⊢ (𝑎 = 𝑏 → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
14	13	adantl 487	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎 = 𝑏) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
15		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → (𝐹‘𝑎) = (𝐹‘𝑏))
16	15	eqcomd 2771	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → (𝐹‘𝑏) = (𝐹‘𝑎))
17		csbeq1 3857	. . . . . . . . . 10 ⊢ ((𝐹‘𝑎) = (𝐹‘𝑏) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑏) / 𝑧⦌𝑆)
18	17	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑏) / 𝑧⦌𝑆)
19	18	breqd 5122	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎 ↔ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎))
20	19	biimpa 482	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎)
21	16, 20	jca 521	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → ((𝐹‘𝑏) = (𝐹‘𝑎) ∧ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎))
22	21	olcd 888	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → ((𝐹‘𝑏)𝑅(𝐹‘𝑎) ∨ ((𝐹‘𝑏) = (𝐹‘𝑎) ∧ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎)))
23	2, 3	fnwe2val 43834	. . . . 5 ⊢ (𝑏𝑇𝑎 ↔ ((𝐹‘𝑏)𝑅(𝐹‘𝑎) ∨ ((𝐹‘𝑏) = (𝐹‘𝑎) ∧ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎)))
24	22, 23	sylibr 237	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → 𝑏𝑇𝑎)
25	24	3mix3d 1357	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
26		fnwe2lem3.a	. . . . . . 7 ⊢ (𝜑 → 𝑎 ∈ 𝐴)
27		fnwe2.s	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
28	2, 3, 27	fnwe2lem1 43835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
29	26, 28	mpdan 700	. . . . . 6 ⊢ (𝜑 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
30		weso 5654	. . . . . 6 ⊢ (⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 Or {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
31	29, 30	syl 18	. . . . 5 ⊢ (𝜑 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 Or {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
32	31	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 Or {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
33		fveqeq2 6894	. . . . 5 ⊢ (𝑦 = 𝑎 → ((𝐹‘𝑦) = (𝐹‘𝑎) ↔ (𝐹‘𝑎) = (𝐹‘𝑎)))
34	26	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑎 ∈ 𝐴)
35		eqidd 2766	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑎) = (𝐹‘𝑎))
36	33, 34, 35	elrabd 3654	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑎 ∈ {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
37		fveqeq2 6894	. . . . 5 ⊢ (𝑦 = 𝑏 → ((𝐹‘𝑦) = (𝐹‘𝑎) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
38		fnwe2lem3.b	. . . . . 6 ⊢ (𝜑 → 𝑏 ∈ 𝐴)
39	38	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑏 ∈ 𝐴)
40		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑎) = (𝐹‘𝑏))
41	40	eqcomd 2771	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑏) = (𝐹‘𝑎))
42	37, 39, 41	elrabd 3654	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑏 ∈ {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
43		solin 5598	. . . 4 ⊢ ((⦋(𝐹‘𝑎) / 𝑧⦌𝑆 Or {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} ∧ (𝑎 ∈ {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} ∧ 𝑏 ∈ {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})) → (𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎))
44	32, 36, 42, 43	syl12anc 850	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎))
45	12, 14, 25, 44	mpjao3dan 1459	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
46		animorrl 996	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑏)𝑅(𝐹‘𝑎)) → ((𝐹‘𝑏)𝑅(𝐹‘𝑎) ∨ ((𝐹‘𝑏) = (𝐹‘𝑎) ∧ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎)))
47	46, 23	sylibr 237	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑏)𝑅(𝐹‘𝑎)) → 𝑏𝑇𝑎)
48	47	3mix3d 1357	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑏)𝑅(𝐹‘𝑎)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
49		fnwe2.r	. . . 4 ⊢ (𝜑 → 𝑅 We 𝐵)
50		weso 5654	. . . 4 ⊢ (𝑅 We 𝐵 → 𝑅 Or 𝐵)
51	49, 50	syl 18	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐵)
52	26	fvresd 6905	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴)‘𝑎) = (𝐹‘𝑎))
53		fnwe2.f	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵)
54	53, 26	ffvelcdmd 7084	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴)‘𝑎) ∈ 𝐵)
55	52, 54	eqeltrrd 2866	. . 3 ⊢ (𝜑 → (𝐹‘𝑎) ∈ 𝐵)
56	38	fvresd 6905	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴)‘𝑏) = (𝐹‘𝑏))
57	53, 38	ffvelcdmd 7084	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴)‘𝑏) ∈ 𝐵)
58	56, 57	eqeltrrd 2866	. . 3 ⊢ (𝜑 → (𝐹‘𝑏) ∈ 𝐵)
59		solin 5598	. . 3 ⊢ ((𝑅 Or 𝐵 ∧ ((𝐹‘𝑎) ∈ 𝐵 ∧ (𝐹‘𝑏) ∈ 𝐵)) → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ (𝐹‘𝑎) = (𝐹‘𝑏) ∨ (𝐹‘𝑏)𝑅(𝐹‘𝑎)))
60	51, 55, 58, 59	syl12anc 850	. 2 ⊢ (𝜑 → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ (𝐹‘𝑎) = (𝐹‘𝑏) ∨ (𝐹‘𝑏)𝑅(𝐹‘𝑎)))
61	6, 45, 48, 60	mpjao3dan 1459	1 ⊢ (𝜑 → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   = wceq 1570   ∈ wcel 2146  {crab 3418  ⦋csb 3854   class class class wbr 5111  {copab 5175   Or wor 5570   We wwe 5615   ↾ cres 5665  ⟶wf 6536  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548

This theorem is used by:  fnwe2  43838