Theorem fnwe2lem3 39137

Description: Lemma for fnwe2 39138. 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 975	. . . 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 39134	. . . 4 ⊢ (𝑎𝑇𝑏 ↔ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))
5	1, 4	sylibr 235	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑎)𝑅(𝐹‘𝑏)) → 𝑎𝑇𝑏)
6	5	3mix1d 1329	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑎)𝑅(𝐹‘𝑏)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
7		simplr 765	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → (𝐹‘𝑎) = (𝐹‘𝑏))
8		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)
9	7, 8	jca 512	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏))
10	9	olcd 871	. . . . 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 235	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → 𝑎𝑇𝑏)
12	11	3mix1d 1329	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
13		3mix2 1324	. . . 4 ⊢ (𝑎 = 𝑏 → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
14	13	adantl 482	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑎 = 𝑏) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
15		simplr 765	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → (𝐹‘𝑎) = (𝐹‘𝑏))
16	15	eqcomd 2801	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → (𝐹‘𝑏) = (𝐹‘𝑎))
17		csbeq1 3814	. . . . . . . . . 10 ⊢ ((𝐹‘𝑎) = (𝐹‘𝑏) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑏) / 𝑧⦌𝑆)
18	17	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑏) / 𝑧⦌𝑆)
19	18	breqd 4973	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎 ↔ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎))
20	19	biimpa 477	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎)
21	16, 20	jca 512	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → ((𝐹‘𝑏) = (𝐹‘𝑎) ∧ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎))
22	21	olcd 871	. . . . 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 39134	. . . . 5 ⊢ (𝑏𝑇𝑎 ↔ ((𝐹‘𝑏)𝑅(𝐹‘𝑎) ∨ ((𝐹‘𝑏) = (𝐹‘𝑎) ∧ 𝑏⦋(𝐹‘𝑏) / 𝑧⦌𝑆𝑎)))
24	22, 23	sylibr 235	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎) → 𝑏𝑇𝑎)
25	24	3mix3d 1331	. . 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 39135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
29	26, 28	mpdan 683	. . . . . 6 ⊢ (𝜑 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
30		weso 5434	. . . . . 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 17	. . . . 5 ⊢ (𝜑 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 Or {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
32	31	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 Or {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
33		fveqeq2 6547	. . . . 5 ⊢ (𝑦 = 𝑎 → ((𝐹‘𝑦) = (𝐹‘𝑎) ↔ (𝐹‘𝑎) = (𝐹‘𝑎)))
34	26	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑎 ∈ 𝐴)
35		eqidd 2796	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑎) = (𝐹‘𝑎))
36	33, 34, 35	elrabd 3620	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑎 ∈ {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
37		fveqeq2 6547	. . . . 5 ⊢ (𝑦 = 𝑏 → ((𝐹‘𝑦) = (𝐹‘𝑎) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
38		fnwe2lem3.b	. . . . . 6 ⊢ (𝜑 → 𝑏 ∈ 𝐴)
39	38	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑏 ∈ 𝐴)
40		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑎) = (𝐹‘𝑏))
41	40	eqcomd 2801	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑏) = (𝐹‘𝑎))
42	37, 39, 41	elrabd 3620	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑏 ∈ {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
43		solin 5386	. . . 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 833	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑎))
45	12, 14, 25, 44	mpjao3dan 1424	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
46		animorrl 975	. . . 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 235	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑏)𝑅(𝐹‘𝑎)) → 𝑏𝑇𝑎)
48	47	3mix3d 1331	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑏)𝑅(𝐹‘𝑎)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
49		fnwe2.r	. . . 4 ⊢ (𝜑 → 𝑅 We 𝐵)
50		weso 5434	. . . 4 ⊢ (𝑅 We 𝐵 → 𝑅 Or 𝐵)
51	49, 50	syl 17	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐵)
52	26	fvresd 6558	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴)‘𝑎) = (𝐹‘𝑎))
53		fnwe2.f	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵)
54	53, 26	ffvelrnd 6717	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴)‘𝑎) ∈ 𝐵)
55	52, 54	eqeltrrd 2884	. . 3 ⊢ (𝜑 → (𝐹‘𝑎) ∈ 𝐵)
56	38	fvresd 6558	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴)‘𝑏) = (𝐹‘𝑏))
57	53, 38	ffvelrnd 6717	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴)‘𝑏) ∈ 𝐵)
58	56, 57	eqeltrrd 2884	. . 3 ⊢ (𝜑 → (𝐹‘𝑏) ∈ 𝐵)
59		solin 5386	. . 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 833	. 2 ⊢ (𝜑 → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ (𝐹‘𝑎) = (𝐹‘𝑏) ∨ (𝐹‘𝑏)𝑅(𝐹‘𝑎)))
61	6, 45, 48, 60	mpjao3dan 1424	1 ⊢ (𝜑 → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 842   ∨ w3o 1079   = wceq 1522   ∈ wcel 2081  {crab 3109  ⦋csb 3811   class class class wbr 4962  {copab 5024   Or wor 5361   We wwe 5401   ↾ cres 5445  ⟶wf 6221  ‘cfv 6225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-id 5348  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233

This theorem is referenced by:  fnwe2  39138