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Theorem pofun 4103
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 = 𝑦𝑋 = 𝑌)
Assertion
Ref Expression
pofun ((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) → 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.
StepHypRef Expression
1 nfcsb1v 2949 . . . . . . 7 𝑥𝑣 / 𝑥𝑋
21nfel1 2233 . . . . . 6 𝑥𝑣 / 𝑥𝑋𝐵
3 csbeq1a 2927 . . . . . . 7 (𝑥 = 𝑣𝑋 = 𝑣 / 𝑥𝑋)
43eleq1d 2151 . . . . . 6 (𝑥 = 𝑣 → (𝑋𝐵𝑣 / 𝑥𝑋𝐵))
52, 4rspc 2706 . . . . 5 (𝑣𝐴 → (∀𝑥𝐴 𝑋𝐵𝑣 / 𝑥𝑋𝐵))
65impcom 123 . . . 4 ((∀𝑥𝐴 𝑋𝐵𝑣𝐴) → 𝑣 / 𝑥𝑋𝐵)
7 poirr 4098 . . . . 5 ((𝑅 Po 𝐵𝑣 / 𝑥𝑋𝐵) → ¬ 𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋)
8 df-br 3812 . . . . . 6 (𝑣𝑆𝑣 ↔ ⟨𝑣, 𝑣⟩ ∈ 𝑆)
9 pofun.1 . . . . . . 7 𝑆 = {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌}
109eleq2i 2149 . . . . . 6 (⟨𝑣, 𝑣⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
11 nfcv 2223 . . . . . . . 8 𝑥𝑅
12 nfcv 2223 . . . . . . . 8 𝑥𝑌
131, 11, 12nfbr 3855 . . . . . . 7 𝑥𝑣 / 𝑥𝑋𝑅𝑌
14 nfv 1462 . . . . . . 7 𝑦𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋
15 vex 2615 . . . . . . 7 𝑣 ∈ V
163breq1d 3821 . . . . . . 7 (𝑥 = 𝑣 → (𝑋𝑅𝑌𝑣 / 𝑥𝑋𝑅𝑌))
17 vex 2615 . . . . . . . . . 10 𝑦 ∈ V
18 pofun.2 . . . . . . . . . 10 (𝑥 = 𝑦𝑋 = 𝑌)
1917, 12, 18csbief 2958 . . . . . . . . 9 𝑦 / 𝑥𝑋 = 𝑌
20 csbeq1 2922 . . . . . . . . 9 (𝑦 = 𝑣𝑦 / 𝑥𝑋 = 𝑣 / 𝑥𝑋)
2119, 20syl5eqr 2129 . . . . . . . 8 (𝑦 = 𝑣𝑌 = 𝑣 / 𝑥𝑋)
2221breq2d 3823 . . . . . . 7 (𝑦 = 𝑣 → (𝑣 / 𝑥𝑋𝑅𝑌𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋))
2313, 14, 15, 15, 16, 22opelopabf 4065 . . . . . 6 (⟨𝑣, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ 𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋)
248, 10, 233bitri 204 . . . . 5 (𝑣𝑆𝑣𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋)
257, 24sylnibr 635 . . . 4 ((𝑅 Po 𝐵𝑣 / 𝑥𝑋𝐵) → ¬ 𝑣𝑆𝑣)
266, 25sylan2 280 . . 3 ((𝑅 Po 𝐵 ∧ (∀𝑥𝐴 𝑋𝐵𝑣𝐴)) → ¬ 𝑣𝑆𝑣)
2726anassrs 392 . 2 (((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) ∧ 𝑣𝐴) → ¬ 𝑣𝑆𝑣)
285com12 30 . . . . . 6 (∀𝑥𝐴 𝑋𝐵 → (𝑣𝐴𝑣 / 𝑥𝑋𝐵))
29 nfcsb1v 2949 . . . . . . . . 9 𝑥𝑤 / 𝑥𝑋
3029nfel1 2233 . . . . . . . 8 𝑥𝑤 / 𝑥𝑋𝐵
31 csbeq1a 2927 . . . . . . . . 9 (𝑥 = 𝑤𝑋 = 𝑤 / 𝑥𝑋)
3231eleq1d 2151 . . . . . . . 8 (𝑥 = 𝑤 → (𝑋𝐵𝑤 / 𝑥𝑋𝐵))
3330, 32rspc 2706 . . . . . . 7 (𝑤𝐴 → (∀𝑥𝐴 𝑋𝐵𝑤 / 𝑥𝑋𝐵))
3433com12 30 . . . . . 6 (∀𝑥𝐴 𝑋𝐵 → (𝑤𝐴𝑤 / 𝑥𝑋𝐵))
35 nfcsb1v 2949 . . . . . . . . 9 𝑥𝑧 / 𝑥𝑋
3635nfel1 2233 . . . . . . . 8 𝑥𝑧 / 𝑥𝑋𝐵
37 csbeq1a 2927 . . . . . . . . 9 (𝑥 = 𝑧𝑋 = 𝑧 / 𝑥𝑋)
3837eleq1d 2151 . . . . . . . 8 (𝑥 = 𝑧 → (𝑋𝐵𝑧 / 𝑥𝑋𝐵))
3936, 38rspc 2706 . . . . . . 7 (𝑧𝐴 → (∀𝑥𝐴 𝑋𝐵𝑧 / 𝑥𝑋𝐵))
4039com12 30 . . . . . 6 (∀𝑥𝐴 𝑋𝐵 → (𝑧𝐴𝑧 / 𝑥𝑋𝐵))
4128, 34, 403anim123d 1251 . . . . 5 (∀𝑥𝐴 𝑋𝐵 → ((𝑣𝐴𝑤𝐴𝑧𝐴) → (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵)))
4241imp 122 . . . 4 ((∀𝑥𝐴 𝑋𝐵 ∧ (𝑣𝐴𝑤𝐴𝑧𝐴)) → (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵))
4342adantll 460 . . 3 (((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) ∧ (𝑣𝐴𝑤𝐴𝑧𝐴)) → (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵))
44 potr 4099 . . . . 5 ((𝑅 Po 𝐵 ∧ (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵)) → ((𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋) → 𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋))
45 df-br 3812 . . . . . . 7 (𝑣𝑆𝑤 ↔ ⟨𝑣, 𝑤⟩ ∈ 𝑆)
469eleq2i 2149 . . . . . . 7 (⟨𝑣, 𝑤⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
47 nfv 1462 . . . . . . . 8 𝑦𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋
48 vex 2615 . . . . . . . 8 𝑤 ∈ V
49 csbeq1 2922 . . . . . . . . . 10 (𝑦 = 𝑤𝑦 / 𝑥𝑋 = 𝑤 / 𝑥𝑋)
5019, 49syl5eqr 2129 . . . . . . . . 9 (𝑦 = 𝑤𝑌 = 𝑤 / 𝑥𝑋)
5150breq2d 3823 . . . . . . . 8 (𝑦 = 𝑤 → (𝑣 / 𝑥𝑋𝑅𝑌𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋))
5213, 47, 15, 48, 16, 51opelopabf 4065 . . . . . . 7 (⟨𝑣, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ 𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋)
5345, 46, 523bitri 204 . . . . . 6 (𝑣𝑆𝑤𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋)
54 df-br 3812 . . . . . . 7 (𝑤𝑆𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑆)
559eleq2i 2149 . . . . . . 7 (⟨𝑤, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
5629, 11, 12nfbr 3855 . . . . . . . 8 𝑥𝑤 / 𝑥𝑋𝑅𝑌
57 nfv 1462 . . . . . . . 8 𝑦𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋
58 vex 2615 . . . . . . . 8 𝑧 ∈ V
5931breq1d 3821 . . . . . . . 8 (𝑥 = 𝑤 → (𝑋𝑅𝑌𝑤 / 𝑥𝑋𝑅𝑌))
60 csbeq1 2922 . . . . . . . . . 10 (𝑦 = 𝑧𝑦 / 𝑥𝑋 = 𝑧 / 𝑥𝑋)
6119, 60syl5eqr 2129 . . . . . . . . 9 (𝑦 = 𝑧𝑌 = 𝑧 / 𝑥𝑋)
6261breq2d 3823 . . . . . . . 8 (𝑦 = 𝑧 → (𝑤 / 𝑥𝑋𝑅𝑌𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋))
6356, 57, 48, 58, 59, 62opelopabf 4065 . . . . . . 7 (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ 𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋)
6454, 55, 633bitri 204 . . . . . 6 (𝑤𝑆𝑧𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋)
6553, 64anbi12i 448 . . . . 5 ((𝑣𝑆𝑤𝑤𝑆𝑧) ↔ (𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋))
66 df-br 3812 . . . . . 6 (𝑣𝑆𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ 𝑆)
679eleq2i 2149 . . . . . 6 (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
68 nfv 1462 . . . . . . 7 𝑦𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋
6961breq2d 3823 . . . . . . 7 (𝑦 = 𝑧 → (𝑣 / 𝑥𝑋𝑅𝑌𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋))
7013, 68, 15, 58, 16, 69opelopabf 4065 . . . . . 6 (⟨𝑣, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ 𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋)
7166, 67, 703bitri 204 . . . . 5 (𝑣𝑆𝑧𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋)
7244, 65, 713imtr4g 203 . . . 4 ((𝑅 Po 𝐵 ∧ (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵)) → ((𝑣𝑆𝑤𝑤𝑆𝑧) → 𝑣𝑆𝑧))
7372adantlr 461 . . 3 (((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) ∧ (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵)) → ((𝑣𝑆𝑤𝑤𝑆𝑧) → 𝑣𝑆𝑧))
7443, 73syldan 276 . 2 (((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) ∧ (𝑣𝐴𝑤𝐴𝑧𝐴)) → ((𝑣𝑆𝑤𝑤𝑆𝑧) → 𝑣𝑆𝑧))
7527, 74ispod 4095 1 ((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) → 𝑆 Po 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  wral 2353  csb 2919  cop 3425   class class class wbr 3811  {copab 3864   Po wpo 4085
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-csb 2920  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-po 4087
This theorem is referenced by: (None)
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