Theorem fnwe2lem3 38407

Description: Lemma for fnwe2 38408. 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		orc 894	. . . . 5 ⊢ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))
2	1	adantl 474	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑎)𝑅(𝐹‘𝑏)) → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))
3		fnwe2.su	. . . . 5 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)
4		fnwe2.t	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}
5	3, 4	fnwe2val 38404	. . . 4 ⊢ (𝑎𝑇𝑏 ↔ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))
6	2, 5	sylibr 226	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑎)𝑅(𝐹‘𝑏)) → 𝑎𝑇𝑏)
7	6	3mix1d 1436	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑎)𝑅(𝐹‘𝑏)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
8		simplr 786	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → (𝐹‘𝑎) = (𝐹‘𝑏))
9		simpr 478	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)
10	8, 9	jca 508	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏))
11	10	olcd 901	. . . . 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)))
12	11, 5	sylibr 226	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → 𝑎𝑇𝑏)
13	12	3mix1d 1436	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
14		3mix2 1431	. . . 4 ⊢ (𝑎 = 𝑏 → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
15	14	adantl 474	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎 = 𝑏) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
16		simplr 786	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → (𝐹‘𝑎) = (𝐹‘𝑏))
17	16	eqcomd 2805	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → (𝐹‘𝑏) = (𝐹‘𝑎))
18		csbeq1 3731	. . . . . . . . . 10 ⊢ ((𝐹‘𝑎) = (𝐹‘𝑏) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑏) / 𝑧⦌𝑆)
19	18	adantl 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑏) / 𝑧⦌𝑆)
20	19	breqd 4854	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎 ↔ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎))
21	20	biimpa 469	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎)
22	17, 21	jca 508	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → ((𝐹‘𝑏) = (𝐹‘𝑎) ∧ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎))
23	22	olcd 901	. . . . 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)))
24	3, 4	fnwe2val 38404	. . . . 5 ⊢ (𝑏𝑇𝑎 ↔ ((𝐹‘𝑏)𝑅(𝐹‘𝑎) ∨ ((𝐹‘𝑏) = (𝐹‘𝑎) ∧ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎)))
25	23, 24	sylibr 226	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → 𝑏𝑇𝑎)
26	25	3mix3d 1438	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
27		fnwe2lem3.a	. . . . . . 7 ⊢ (𝜑 → 𝑎 ∈ 𝐴)
28		fnwe2.s	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
29	3, 4, 28	fnwe2lem1 38405	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
30	27, 29	mpdan 679	. . . . . 6 ⊢ (𝜑 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
31		weso 5303	. . . . . 6 ⊢ (⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 Or {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
32	30, 31	syl 17	. . . . 5 ⊢ (𝜑 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 Or {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
33	32	adantr 473	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 Or {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
34	27	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑎 ∈ 𝐴)
35		eqidd 2800	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑎) = (𝐹‘𝑎))
36		fveqeq2 6420	. . . . . 6 ⊢ (𝑦 = 𝑎 → ((𝐹‘𝑦) = (𝐹‘𝑎) ↔ (𝐹‘𝑎) = (𝐹‘𝑎)))
37	36	elrab 3556	. . . . 5 ⊢ (𝑎 ∈ {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} ↔ (𝑎 ∈ 𝐴 ∧ (𝐹‘𝑎) = (𝐹‘𝑎)))
38	34, 35, 37	sylanbrc 579	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑎 ∈ {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
39		fnwe2lem3.b	. . . . . 6 ⊢ (𝜑 → 𝑏 ∈ 𝐴)
40	39	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑏 ∈ 𝐴)
41		simpr 478	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑎) = (𝐹‘𝑏))
42	41	eqcomd 2805	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑏) = (𝐹‘𝑎))
43		fveqeq2 6420	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝐹‘𝑦) = (𝐹‘𝑎) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
44	43	elrab 3556	. . . . 5 ⊢ (𝑏 ∈ {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} ↔ (𝑏 ∈ 𝐴 ∧ (𝐹‘𝑏) = (𝐹‘𝑎)))
45	40, 42, 44	sylanbrc 579	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑏 ∈ {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
46		solin 5256	. . . 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))
47	33, 38, 45, 46	syl12anc 866	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎))
48	13, 15, 26, 47	mpjao3dan 1557	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
49		orc 894	. . . . 5 ⊢ ((𝐹‘𝑏)𝑅(𝐹‘𝑎) → ((𝐹‘𝑏)𝑅(𝐹‘𝑎) ∨ ((𝐹‘𝑏) = (𝐹‘𝑎) ∧ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎)))
50	49	adantl 474	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑏)𝑅(𝐹‘𝑎)) → ((𝐹‘𝑏)𝑅(𝐹‘𝑎) ∨ ((𝐹‘𝑏) = (𝐹‘𝑎) ∧ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎)))
51	50, 24	sylibr 226	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑏)𝑅(𝐹‘𝑎)) → 𝑏𝑇𝑎)
52	51	3mix3d 1438	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑏)𝑅(𝐹‘𝑎)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
53		fnwe2.r	. . . 4 ⊢ (𝜑 → 𝑅 We 𝐵)
54		weso 5303	. . . 4 ⊢ (𝑅 We 𝐵 → 𝑅 Or 𝐵)
55	53, 54	syl 17	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐵)
56		fvres 6430	. . . . 5 ⊢ (𝑎 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑎) = (𝐹‘𝑎))
57	27, 56	syl 17	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴)‘𝑎) = (𝐹‘𝑎))
58		fnwe2.f	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵)
59	58, 27	ffvelrnd 6586	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴)‘𝑎) ∈ 𝐵)
60	57, 59	eqeltrrd 2879	. . 3 ⊢ (𝜑 → (𝐹‘𝑎) ∈ 𝐵)
61		fvres 6430	. . . . 5 ⊢ (𝑏 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑏) = (𝐹‘𝑏))
62	39, 61	syl 17	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴)‘𝑏) = (𝐹‘𝑏))
63	58, 39	ffvelrnd 6586	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴)‘𝑏) ∈ 𝐵)
64	62, 63	eqeltrrd 2879	. . 3 ⊢ (𝜑 → (𝐹‘𝑏) ∈ 𝐵)
65		solin 5256	. . 3 ⊢ ((𝑅 Or 𝐵 ∧ ((𝐹‘𝑎) ∈ 𝐵 ∧ (𝐹‘𝑏) ∈ 𝐵)) → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ (𝐹‘𝑎) = (𝐹‘𝑏) ∨ (𝐹‘𝑏)𝑅(𝐹‘𝑎)))
66	55, 60, 64, 65	syl12anc 866	. 2 ⊢ (𝜑 → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ (𝐹‘𝑎) = (𝐹‘𝑏) ∨ (𝐹‘𝑏)𝑅(𝐹‘𝑎)))
67	7, 48, 52, 66	mpjao3dan 1557	1 ⊢ (𝜑 → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))

Syntax hints:   → wi 4   ∧ wa 385   ∨ wo 874   ∨ w3o 1107   = wceq 1653   ∈ wcel 2157  {crab 3093  ⦋csb 3728   class class class wbr 4843  {copab 4905   Or wor 5232   We wwe 5270   ↾ cres 5314  ⟶wf 6097  ‘cfv 6101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109

This theorem is referenced by:  fnwe2  38408