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Theorem pofun 4130
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 2961 . . . . . . 7 𝑥𝑣 / 𝑥𝑋
21nfel1 2239 . . . . . 6 𝑥𝑣 / 𝑥𝑋𝐵
3 csbeq1a 2939 . . . . . . 7 (𝑥 = 𝑣𝑋 = 𝑣 / 𝑥𝑋)
43eleq1d 2156 . . . . . 6 (𝑥 = 𝑣 → (𝑋𝐵𝑣 / 𝑥𝑋𝐵))
52, 4rspc 2716 . . . . 5 (𝑣𝐴 → (∀𝑥𝐴 𝑋𝐵𝑣 / 𝑥𝑋𝐵))
65impcom 123 . . . 4 ((∀𝑥𝐴 𝑋𝐵𝑣𝐴) → 𝑣 / 𝑥𝑋𝐵)
7 poirr 4125 . . . . 5 ((𝑅 Po 𝐵𝑣 / 𝑥𝑋𝐵) → ¬ 𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋)
8 df-br 3838 . . . . . 6 (𝑣𝑆𝑣 ↔ ⟨𝑣, 𝑣⟩ ∈ 𝑆)
9 pofun.1 . . . . . . 7 𝑆 = {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌}
109eleq2i 2154 . . . . . 6 (⟨𝑣, 𝑣⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
11 nfcv 2228 . . . . . . . 8 𝑥𝑅
12 nfcv 2228 . . . . . . . 8 𝑥𝑌
131, 11, 12nfbr 3881 . . . . . . 7 𝑥𝑣 / 𝑥𝑋𝑅𝑌
14 nfv 1466 . . . . . . 7 𝑦𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋
15 vex 2622 . . . . . . 7 𝑣 ∈ V
163breq1d 3847 . . . . . . 7 (𝑥 = 𝑣 → (𝑋𝑅𝑌𝑣 / 𝑥𝑋𝑅𝑌))
17 vex 2622 . . . . . . . . . 10 𝑦 ∈ V
18 pofun.2 . . . . . . . . . 10 (𝑥 = 𝑦𝑋 = 𝑌)
1917, 12, 18csbief 2970 . . . . . . . . 9 𝑦 / 𝑥𝑋 = 𝑌
20 csbeq1 2934 . . . . . . . . 9 (𝑦 = 𝑣𝑦 / 𝑥𝑋 = 𝑣 / 𝑥𝑋)
2119, 20syl5eqr 2134 . . . . . . . 8 (𝑦 = 𝑣𝑌 = 𝑣 / 𝑥𝑋)
2221breq2d 3849 . . . . . . 7 (𝑦 = 𝑣 → (𝑣 / 𝑥𝑋𝑅𝑌𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋))
2313, 14, 15, 15, 16, 22opelopabf 4092 . . . . . 6 (⟨𝑣, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ 𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋)
248, 10, 233bitri 204 . . . . 5 (𝑣𝑆𝑣𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋)
257, 24sylnibr 637 . . . 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 2961 . . . . . . . . 9 𝑥𝑤 / 𝑥𝑋
3029nfel1 2239 . . . . . . . 8 𝑥𝑤 / 𝑥𝑋𝐵
31 csbeq1a 2939 . . . . . . . . 9 (𝑥 = 𝑤𝑋 = 𝑤 / 𝑥𝑋)
3231eleq1d 2156 . . . . . . . 8 (𝑥 = 𝑤 → (𝑋𝐵𝑤 / 𝑥𝑋𝐵))
3330, 32rspc 2716 . . . . . . 7 (𝑤𝐴 → (∀𝑥𝐴 𝑋𝐵𝑤 / 𝑥𝑋𝐵))
3433com12 30 . . . . . 6 (∀𝑥𝐴 𝑋𝐵 → (𝑤𝐴𝑤 / 𝑥𝑋𝐵))
35 nfcsb1v 2961 . . . . . . . . 9 𝑥𝑧 / 𝑥𝑋
3635nfel1 2239 . . . . . . . 8 𝑥𝑧 / 𝑥𝑋𝐵
37 csbeq1a 2939 . . . . . . . . 9 (𝑥 = 𝑧𝑋 = 𝑧 / 𝑥𝑋)
3837eleq1d 2156 . . . . . . . 8 (𝑥 = 𝑧 → (𝑋𝐵𝑧 / 𝑥𝑋𝐵))
3936, 38rspc 2716 . . . . . . 7 (𝑧𝐴 → (∀𝑥𝐴 𝑋𝐵𝑧 / 𝑥𝑋𝐵))
4039com12 30 . . . . . 6 (∀𝑥𝐴 𝑋𝐵 → (𝑧𝐴𝑧 / 𝑥𝑋𝐵))
4128, 34, 403anim123d 1255 . . . . 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 4126 . . . . 5 ((𝑅 Po 𝐵 ∧ (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵)) → ((𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋) → 𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋))
45 df-br 3838 . . . . . . 7 (𝑣𝑆𝑤 ↔ ⟨𝑣, 𝑤⟩ ∈ 𝑆)
469eleq2i 2154 . . . . . . 7 (⟨𝑣, 𝑤⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
47 nfv 1466 . . . . . . . 8 𝑦𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋
48 vex 2622 . . . . . . . 8 𝑤 ∈ V
49 csbeq1 2934 . . . . . . . . . 10 (𝑦 = 𝑤𝑦 / 𝑥𝑋 = 𝑤 / 𝑥𝑋)
5019, 49syl5eqr 2134 . . . . . . . . 9 (𝑦 = 𝑤𝑌 = 𝑤 / 𝑥𝑋)
5150breq2d 3849 . . . . . . . 8 (𝑦 = 𝑤 → (𝑣 / 𝑥𝑋𝑅𝑌𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋))
5213, 47, 15, 48, 16, 51opelopabf 4092 . . . . . . 7 (⟨𝑣, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ 𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋)
5345, 46, 523bitri 204 . . . . . 6 (𝑣𝑆𝑤𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋)
54 df-br 3838 . . . . . . 7 (𝑤𝑆𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑆)
559eleq2i 2154 . . . . . . 7 (⟨𝑤, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
5629, 11, 12nfbr 3881 . . . . . . . 8 𝑥𝑤 / 𝑥𝑋𝑅𝑌
57 nfv 1466 . . . . . . . 8 𝑦𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋
58 vex 2622 . . . . . . . 8 𝑧 ∈ V
5931breq1d 3847 . . . . . . . 8 (𝑥 = 𝑤 → (𝑋𝑅𝑌𝑤 / 𝑥𝑋𝑅𝑌))
60 csbeq1 2934 . . . . . . . . . 10 (𝑦 = 𝑧𝑦 / 𝑥𝑋 = 𝑧 / 𝑥𝑋)
6119, 60syl5eqr 2134 . . . . . . . . 9 (𝑦 = 𝑧𝑌 = 𝑧 / 𝑥𝑋)
6261breq2d 3849 . . . . . . . 8 (𝑦 = 𝑧 → (𝑤 / 𝑥𝑋𝑅𝑌𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋))
6356, 57, 48, 58, 59, 62opelopabf 4092 . . . . . . 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 3838 . . . . . 6 (𝑣𝑆𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ 𝑆)
679eleq2i 2154 . . . . . 6 (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
68 nfv 1466 . . . . . . 7 𝑦𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋
6961breq2d 3849 . . . . . . 7 (𝑦 = 𝑧 → (𝑣 / 𝑥𝑋𝑅𝑌𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋))
7013, 68, 15, 58, 16, 69opelopabf 4092 . . . . . 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 4122 1 ((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) → 𝑆 Po 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  w3a 924   = wceq 1289  wcel 1438  wral 2359  csb 2931  cop 3444   class class class wbr 3837  {copab 3890   Po wpo 4112
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-po 4114
This theorem is referenced by: (None)
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