Theorem pofun 4067

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 2938	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑣 / 𝑥⦌𝑋
2	1	nfel1 2229	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵
3		csbeq1a 2916	. . . . . . 7 ⊢ (𝑥 = 𝑣 → 𝑋 = ⦋𝑣 / 𝑥⦌𝑋)
4	3	eleq1d 2147	. . . . . 6 ⊢ (𝑥 = 𝑣 → (𝑋 ∈ 𝐵 ↔ ⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵))
5	2, 4	rspc 2695	. . . . 5 ⊢ (𝑣 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 → ⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵))
6	5	impcom 123	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) → ⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵)
7		poirr 4062	. . . . 5 ⊢ ((𝑅 Po 𝐵 ∧ ⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵) → ¬ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑣 / 𝑥⦌𝑋)
8		df-br 3786	. . . . . 6 ⊢ (𝑣𝑆𝑣 ↔ ⟨𝑣, 𝑣⟩ ∈ 𝑆)
9		pofun.1	. . . . . . 7 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌}
10	9	eleq2i 2145	. . . . . 6 ⊢ (⟨𝑣, 𝑣⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
11		nfcv 2219	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
12		nfcv 2219	. . . . . . . 8 ⊢ Ⅎ𝑥𝑌
13	1, 11, 12	nfbr 3829	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑣 / 𝑥⦌𝑋𝑅𝑌
14		nfv 1461	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑣 / 𝑥⦌𝑋
15		vex 2604	. . . . . . 7 ⊢ 𝑣 ∈ V
16	3	breq1d 3795	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (𝑋𝑅𝑌 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅𝑌))
17		vex 2604	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
18		pofun.2	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑋 = 𝑌)
19	17, 12, 18	csbief 2947	. . . . . . . . 9 ⊢ ⦋𝑦 / 𝑥⦌𝑋 = 𝑌
20		csbeq1 2911	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ⦋𝑦 / 𝑥⦌𝑋 = ⦋𝑣 / 𝑥⦌𝑋)
21	19, 20	syl5eqr 2127	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → 𝑌 = ⦋𝑣 / 𝑥⦌𝑋)
22	21	breq2d 3797	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (⦋𝑣 / 𝑥⦌𝑋𝑅𝑌 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑣 / 𝑥⦌𝑋))
23	13, 14, 15, 15, 16, 22	opelopabf 4029	. . . . . 6 ⊢ (⟨𝑣, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑣 / 𝑥⦌𝑋)
24	8, 10, 23	3bitri 204	. . . . 5 ⊢ (𝑣𝑆𝑣 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑣 / 𝑥⦌𝑋)
25	7, 24	sylnibr 634	. . . 4 ⊢ ((𝑅 Po 𝐵 ∧ ⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵) → ¬ 𝑣𝑆𝑣)
26	6, 25	sylan2 280	. . 3 ⊢ ((𝑅 Po 𝐵 ∧ (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴)) → ¬ 𝑣𝑆𝑣)
27	26	anassrs 392	. 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 2938	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝑋
30	29	nfel1 2229	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵
31		csbeq1a 2916	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → 𝑋 = ⦋𝑤 / 𝑥⦌𝑋)
32	31	eleq1d 2147	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑋 ∈ 𝐵 ↔ ⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵))
33	30, 32	rspc 2695	. . . . . . 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 2938	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑋
36	35	nfel1 2229	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵
37		csbeq1a 2916	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑋 = ⦋𝑧 / 𝑥⦌𝑋)
38	37	eleq1d 2147	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑋 ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵))
39	36, 38	rspc 2695	. . . . . . 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 1250	. . . . 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 122	. . . 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 459	. . 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 4063	. . . . 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 3786	. . . . . . 7 ⊢ (𝑣𝑆𝑤 ↔ ⟨𝑣, 𝑤⟩ ∈ 𝑆)
46	9	eleq2i 2145	. . . . . . 7 ⊢ (⟨𝑣, 𝑤⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
47		nfv 1461	. . . . . . . 8 ⊢ Ⅎ𝑦⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑤 / 𝑥⦌𝑋
48		vex 2604	. . . . . . . 8 ⊢ 𝑤 ∈ V
49		csbeq1 2911	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ⦋𝑦 / 𝑥⦌𝑋 = ⦋𝑤 / 𝑥⦌𝑋)
50	19, 49	syl5eqr 2127	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → 𝑌 = ⦋𝑤 / 𝑥⦌𝑋)
51	50	breq2d 3797	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (⦋𝑣 / 𝑥⦌𝑋𝑅𝑌 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑤 / 𝑥⦌𝑋))
52	13, 47, 15, 48, 16, 51	opelopabf 4029	. . . . . . 7 ⊢ (⟨𝑣, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑤 / 𝑥⦌𝑋)
53	45, 46, 52	3bitri 204	. . . . . 6 ⊢ (𝑣𝑆𝑤 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑤 / 𝑥⦌𝑋)
54		df-br 3786	. . . . . . 7 ⊢ (𝑤𝑆𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑆)
55	9	eleq2i 2145	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
56	29, 11, 12	nfbr 3829	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝑋𝑅𝑌
57		nfv 1461	. . . . . . . 8 ⊢ Ⅎ𝑦⦋𝑤 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋
58		vex 2604	. . . . . . . 8 ⊢ 𝑧 ∈ V
59	31	breq1d 3795	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑋𝑅𝑌 ↔ ⦋𝑤 / 𝑥⦌𝑋𝑅𝑌))
60		csbeq1 2911	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝑋 = ⦋𝑧 / 𝑥⦌𝑋)
61	19, 60	syl5eqr 2127	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → 𝑌 = ⦋𝑧 / 𝑥⦌𝑋)
62	61	breq2d 3797	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (⦋𝑤 / 𝑥⦌𝑋𝑅𝑌 ↔ ⦋𝑤 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋))
63	56, 57, 48, 58, 59, 62	opelopabf 4029	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ ⦋𝑤 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋)
64	54, 55, 63	3bitri 204	. . . . . 6 ⊢ (𝑤𝑆𝑧 ↔ ⦋𝑤 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋)
65	53, 64	anbi12i 447	. . . . 5 ⊢ ((𝑣𝑆𝑤 ∧ 𝑤𝑆𝑧) ↔ (⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑤 / 𝑥⦌𝑋 ∧ ⦋𝑤 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋))
66		df-br 3786	. . . . . 6 ⊢ (𝑣𝑆𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ 𝑆)
67	9	eleq2i 2145	. . . . . 6 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
68		nfv 1461	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋
69	61	breq2d 3797	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (⦋𝑣 / 𝑥⦌𝑋𝑅𝑌 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋))
70	13, 68, 15, 58, 16, 69	opelopabf 4029	. . . . . 6 ⊢ (⟨𝑣, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋)
71	66, 67, 70	3bitri 204	. . . . 5 ⊢ (𝑣𝑆𝑧 ↔ ⦋𝑣 / 𝑥⦌𝑋𝑅⦋𝑧 / 𝑥⦌𝑋)
72	44, 65, 71	3imtr4g 203	. . . 4 ⊢ ((𝑅 Po 𝐵 ∧ (⦋𝑣 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑤 / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝑋 ∈ 𝐵)) → ((𝑣𝑆𝑤 ∧ 𝑤𝑆𝑧) → 𝑣𝑆𝑧))
73	72	adantlr 460	. . 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 276	. 2 ⊢ (((𝑅 Po 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑣𝑆𝑤 ∧ 𝑤𝑆𝑧) → 𝑣𝑆𝑧))
75	27, 74	ispod 4059	1 ⊢ ((𝑅 Po 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵) → 𝑆 Po 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ⦋csb 2908  ⟨cop 3401   class class class wbr 3785  {copab 3838   Po wpo 4049

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-po 4051

This theorem is referenced by: (None)