Theorem pofun 4347

Description: A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.)

Hypotheses

Ref	Expression
pofun.1	⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌}
pofun.2	⊢ (𝑥 = 𝑦 → 𝑋 = 𝑌)

Assertion

Ref	Expression
pofun	⊢ ((𝑅 Po 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵) → 𝑆 Po 𝐴)

Distinct variable groups:   𝑥,𝑅,𝑦   𝑦,𝑋   𝑥,𝑌   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)   𝑆(𝑥,𝑦)   𝑋(𝑥)   𝑌(𝑦)

Proof of Theorem pofun

Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcsb1v 3117	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑣 / 𝑥⦌𝑋
2	1	nfel1 2350	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵
3		csbeq1a 3093	. . . . . . 7 ⊢ (𝑥 = 𝑣 → 𝑋 = ⦋𝑣 / 𝑥⦌𝑋)
4	3	eleq1d 2265	. . . . . 6 ⊢ (𝑥 = 𝑣 → (𝑋 ∈ 𝐵 ↔ ⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵))
5	2, 4	rspc 2862	. . . . 5 ⊢ (𝑣 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 → ⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵))
6	5	impcom 125	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) → ⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵)
7		poirr 4342	. . . . 5 ⊢ ((𝑅 Po 𝐵 ∧ ⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵) → ¬ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑣 / 𝑥⦌𝑋)
8		df-br 4034	. . . . . 6 ⊢ (𝑣𝑆𝑣 ↔ ⟨𝑣, 𝑣⟩ ∈ 𝑆)
9		pofun.1	. . . . . . 7 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌}
10	9	eleq2i 2263	. . . . . 6 ⊢ (⟨𝑣, 𝑣⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
11		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
12		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥𝑌
13	1, 11, 12	nfbr 4079	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑣 / 𝑥⦌𝑋𝑅𝑌
14		nfv 1542	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑣 / 𝑥⦌𝑋
15		vex 2766	. . . . . . 7 ⊢ 𝑣 ∈ V
16	3	breq1d 4043	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (𝑋𝑅𝑌 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅𝑌))
17		vex 2766	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
18		pofun.2	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑋 = 𝑌)
19	17, 12, 18	csbief 3129	. . . . . . . . 9 ⊢ ⦋𝑦 / 𝑥⦌𝑋 = 𝑌
20		csbeq1 3087	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ⦋𝑦 / 𝑥⦌𝑋 = ⦋𝑣 / 𝑥⦌𝑋)
21	19, 20	eqtr3id 2243	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → 𝑌 = ⦋𝑣 / 𝑥⦌𝑋)
22	21	breq2d 4045	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (⦋𝑣 / 𝑥⦌𝑋𝑅𝑌 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑣 / 𝑥⦌𝑋))
23	13, 14, 15, 15, 16, 22	opelopabf 4309	. . . . . 6 ⊢ (⟨𝑣, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑣 / 𝑥⦌𝑋)
24	8, 10, 23	3bitri 206	. . . . 5 ⊢ (𝑣𝑆𝑣 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑣 / 𝑥⦌𝑋)
25	7, 24	sylnibr 678	. . . 4 ⊢ ((𝑅 Po 𝐵 ∧ ⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵) → ¬ 𝑣𝑆𝑣)
26	6, 25	sylan2 286	. . 3 ⊢ ((𝑅 Po 𝐵 ∧ (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴)) → ¬ 𝑣𝑆𝑣)
27	26	anassrs 400	. 2 ⊢ (((𝑅 Po 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵) ∧ 𝑣 ∈ 𝐴) → ¬ 𝑣𝑆𝑣)
28	5	com12 30	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 → (𝑣 ∈ 𝐴 → ⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵))
29		nfcsb1v 3117	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝑋
30	29	nfel1 2350	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵
31		csbeq1a 3093	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → 𝑋 = ⦋𝑤 / 𝑥⦌𝑋)
32	31	eleq1d 2265	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑋 ∈ 𝐵 ↔ ⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵))
33	30, 32	rspc 2862	. . . . . . 7 ⊢ (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 → ⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵))
34	33	com12 30	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 → (𝑤 ∈ 𝐴 → ⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵))
35		nfcsb1v 3117	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑋
36	35	nfel1 2350	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵
37		csbeq1a 3093	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑋 = ⦋𝑧 / 𝑥⦌𝑋)
38	37	eleq1d 2265	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑋 ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵))
39	36, 38	rspc 2862	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 → ⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵))
40	39	com12 30	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 → (𝑧 ∈ 𝐴 → ⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵))
41	28, 34, 40	3anim123d 1330	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 → ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵)))
42	41	imp 124	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵))
43	42	adantll 476	. . 3 ⊢ (((𝑅 Po 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵))
44		potr 4343	. . . . 5 ⊢ ((𝑅 Po 𝐵 ∧ (⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵)) → ((⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑤 / 𝑥⦌𝑋 ∧ ⦋𝑤 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋) → ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋))
45		df-br 4034	. . . . . . 7 ⊢ (𝑣𝑆𝑤 ↔ ⟨𝑣, 𝑤⟩ ∈ 𝑆)
46	9	eleq2i 2263	. . . . . . 7 ⊢ (⟨𝑣, 𝑤⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
47		nfv 1542	. . . . . . . 8 ⊢ Ⅎ𝑦⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑤 / 𝑥⦌𝑋
48		vex 2766	. . . . . . . 8 ⊢ 𝑤 ∈ V
49		csbeq1 3087	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ⦋𝑦 / 𝑥⦌𝑋 = ⦋𝑤 / 𝑥⦌𝑋)
50	19, 49	eqtr3id 2243	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → 𝑌 = ⦋𝑤 / 𝑥⦌𝑋)
51	50	breq2d 4045	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (⦋𝑣 / 𝑥⦌𝑋𝑅𝑌 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑤 / 𝑥⦌𝑋))
52	13, 47, 15, 48, 16, 51	opelopabf 4309	. . . . . . 7 ⊢ (⟨𝑣, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑤 / 𝑥⦌𝑋)
53	45, 46, 52	3bitri 206	. . . . . 6 ⊢ (𝑣𝑆𝑤 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑤 / 𝑥⦌𝑋)
54		df-br 4034	. . . . . . 7 ⊢ (𝑤𝑆𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑆)
55	9	eleq2i 2263	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
56	29, 11, 12	nfbr 4079	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝑋𝑅𝑌
57		nfv 1542	. . . . . . . 8 ⊢ Ⅎ𝑦⦋𝑤 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋
58		vex 2766	. . . . . . . 8 ⊢ 𝑧 ∈ V
59	31	breq1d 4043	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑋𝑅𝑌 ↔ ⦋𝑤 / 𝑥⦌𝑋𝑅𝑌))
60		csbeq1 3087	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝑋 = ⦋𝑧 / 𝑥⦌𝑋)
61	19, 60	eqtr3id 2243	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → 𝑌 = ⦋𝑧 / 𝑥⦌𝑋)
62	61	breq2d 4045	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (⦋𝑤 / 𝑥⦌𝑋𝑅𝑌 ↔ ⦋𝑤 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋))
63	56, 57, 48, 58, 59, 62	opelopabf 4309	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ ⦋𝑤 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋)
64	54, 55, 63	3bitri 206	. . . . . 6 ⊢ (𝑤𝑆𝑧 ↔ ⦋𝑤 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋)
65	53, 64	anbi12i 460	. . . . 5 ⊢ ((𝑣𝑆𝑤 ∧ 𝑤𝑆𝑧) ↔ (⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑤 / 𝑥⦌𝑋 ∧ ⦋𝑤 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋))
66		df-br 4034	. . . . . 6 ⊢ (𝑣𝑆𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ 𝑆)
67	9	eleq2i 2263	. . . . . 6 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
68		nfv 1542	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋
69	61	breq2d 4045	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (⦋𝑣 / 𝑥⦌𝑋𝑅𝑌 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋))
70	13, 68, 15, 58, 16, 69	opelopabf 4309	. . . . . 6 ⊢ (⟨𝑣, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋)
71	66, 67, 70	3bitri 206	. . . . 5 ⊢ (𝑣𝑆𝑧 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋)
72	44, 65, 71	3imtr4g 205	. . . 4 ⊢ ((𝑅 Po 𝐵 ∧ (⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵)) → ((𝑣𝑆𝑤 ∧ 𝑤𝑆𝑧) → 𝑣𝑆𝑧))
73	72	adantlr 477	. . 3 ⊢ (((𝑅 Po 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵) ∧ (⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵)) → ((𝑣𝑆𝑤 ∧ 𝑤𝑆𝑧) → 𝑣𝑆𝑧))
74	43, 73	syldan 282	. 2 ⊢ (((𝑅 Po 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑣𝑆𝑤 ∧ 𝑤𝑆𝑧) → 𝑣𝑆𝑧))
75	27, 74	ispod 4339	1 ⊢ ((𝑅 Po 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵) → 𝑆 Po 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ⦋csb 3084  ⟨cop 3625   class class class wbr 4033  {copab 4093   Po wpo 4329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-po 4331

This theorem is referenced by: (None)