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Theorem pofun 4040
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 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.
StepHypRef Expression
1 nfcsb1v 2876 . . . . . . 7 xv / x𝑋
21nfel1 2185 . . . . . 6 xv / x𝑋 B
3 csbeq1a 2854 . . . . . . 7 (x = v𝑋 = v / x𝑋)
43eleq1d 2103 . . . . . 6 (x = v → (𝑋 Bv / x𝑋 B))
52, 4rspc 2644 . . . . 5 (v A → (x A 𝑋 Bv / x𝑋 B))
65impcom 116 . . . 4 ((x A 𝑋 B v A) → v / x𝑋 B)
7 poirr 4035 . . . . 5 ((𝑅 Po B v / x𝑋 B) → ¬ v / x𝑋𝑅v / x𝑋)
8 df-br 3756 . . . . . 6 (v𝑆v ↔ ⟨v, v 𝑆)
9 pofun.1 . . . . . . 7 𝑆 = {⟨x, y⟩ ∣ 𝑋𝑅𝑌}
109eleq2i 2101 . . . . . 6 (⟨v, v 𝑆 ↔ ⟨v, v {⟨x, y⟩ ∣ 𝑋𝑅𝑌})
11 nfcv 2175 . . . . . . . 8 x𝑅
12 nfcv 2175 . . . . . . . 8 x𝑌
131, 11, 12nfbr 3799 . . . . . . 7 xv / x𝑋𝑅𝑌
14 nfv 1418 . . . . . . 7 yv / x𝑋𝑅v / x𝑋
15 vex 2554 . . . . . . 7 v V
163breq1d 3765 . . . . . . 7 (x = v → (𝑋𝑅𝑌v / x𝑋𝑅𝑌))
17 vex 2554 . . . . . . . . . 10 y V
18 pofun.2 . . . . . . . . . 10 (x = y𝑋 = 𝑌)
1917, 12, 18csbief 2885 . . . . . . . . 9 y / x𝑋 = 𝑌
20 csbeq1 2849 . . . . . . . . 9 (y = vy / x𝑋 = v / x𝑋)
2119, 20syl5eqr 2083 . . . . . . . 8 (y = v𝑌 = v / x𝑋)
2221breq2d 3767 . . . . . . 7 (y = v → (v / x𝑋𝑅𝑌v / x𝑋𝑅v / x𝑋))
2313, 14, 15, 15, 16, 22opelopabf 4002 . . . . . 6 (⟨v, v {⟨x, y⟩ ∣ 𝑋𝑅𝑌} ↔ v / x𝑋𝑅v / x𝑋)
248, 10, 233bitri 195 . . . . 5 (v𝑆vv / x𝑋𝑅v / x𝑋)
257, 24sylnibr 601 . . . 4 ((𝑅 Po B v / x𝑋 B) → ¬ v𝑆v)
266, 25sylan2 270 . . 3 ((𝑅 Po B (x A 𝑋 B v A)) → ¬ v𝑆v)
2726anassrs 380 . 2 (((𝑅 Po B x A 𝑋 B) v A) → ¬ v𝑆v)
285com12 27 . . . . . 6 (x A 𝑋 B → (v Av / x𝑋 B))
29 nfcsb1v 2876 . . . . . . . . 9 xw / x𝑋
3029nfel1 2185 . . . . . . . 8 xw / x𝑋 B
31 csbeq1a 2854 . . . . . . . . 9 (x = w𝑋 = w / x𝑋)
3231eleq1d 2103 . . . . . . . 8 (x = w → (𝑋 Bw / x𝑋 B))
3330, 32rspc 2644 . . . . . . 7 (w A → (x A 𝑋 Bw / x𝑋 B))
3433com12 27 . . . . . 6 (x A 𝑋 B → (w Aw / x𝑋 B))
35 nfcsb1v 2876 . . . . . . . . 9 xz / x𝑋
3635nfel1 2185 . . . . . . . 8 xz / x𝑋 B
37 csbeq1a 2854 . . . . . . . . 9 (x = z𝑋 = z / x𝑋)
3837eleq1d 2103 . . . . . . . 8 (x = z → (𝑋 Bz / x𝑋 B))
3936, 38rspc 2644 . . . . . . 7 (z A → (x A 𝑋 Bz / x𝑋 B))
4039com12 27 . . . . . 6 (x A 𝑋 B → (z Az / x𝑋 B))
4128, 34, 403anim123d 1213 . . . . 5 (x A 𝑋 B → ((v A w A z A) → (v / x𝑋 B w / x𝑋 B z / x𝑋 B)))
4241imp 115 . . . 4 ((x A 𝑋 B (v A w A z A)) → (v / x𝑋 B w / x𝑋 B z / x𝑋 B))
4342adantll 445 . . 3 (((𝑅 Po B x A 𝑋 B) (v A w A z A)) → (v / x𝑋 B w / x𝑋 B z / x𝑋 B))
44 potr 4036 . . . . 5 ((𝑅 Po B (v / x𝑋 B w / x𝑋 B z / x𝑋 B)) → ((v / x𝑋𝑅w / x𝑋 w / x𝑋𝑅z / x𝑋) → v / x𝑋𝑅z / x𝑋))
45 df-br 3756 . . . . . . 7 (v𝑆w ↔ ⟨v, w 𝑆)
469eleq2i 2101 . . . . . . 7 (⟨v, w 𝑆 ↔ ⟨v, w {⟨x, y⟩ ∣ 𝑋𝑅𝑌})
47 nfv 1418 . . . . . . . 8 yv / x𝑋𝑅w / x𝑋
48 vex 2554 . . . . . . . 8 w V
49 csbeq1 2849 . . . . . . . . . 10 (y = wy / x𝑋 = w / x𝑋)
5019, 49syl5eqr 2083 . . . . . . . . 9 (y = w𝑌 = w / x𝑋)
5150breq2d 3767 . . . . . . . 8 (y = w → (v / x𝑋𝑅𝑌v / x𝑋𝑅w / x𝑋))
5213, 47, 15, 48, 16, 51opelopabf 4002 . . . . . . 7 (⟨v, w {⟨x, y⟩ ∣ 𝑋𝑅𝑌} ↔ v / x𝑋𝑅w / x𝑋)
5345, 46, 523bitri 195 . . . . . 6 (v𝑆wv / x𝑋𝑅w / x𝑋)
54 df-br 3756 . . . . . . 7 (w𝑆z ↔ ⟨w, z 𝑆)
559eleq2i 2101 . . . . . . 7 (⟨w, z 𝑆 ↔ ⟨w, z {⟨x, y⟩ ∣ 𝑋𝑅𝑌})
5629, 11, 12nfbr 3799 . . . . . . . 8 xw / x𝑋𝑅𝑌
57 nfv 1418 . . . . . . . 8 yw / x𝑋𝑅z / x𝑋
58 vex 2554 . . . . . . . 8 z V
5931breq1d 3765 . . . . . . . 8 (x = w → (𝑋𝑅𝑌w / x𝑋𝑅𝑌))
60 csbeq1 2849 . . . . . . . . . 10 (y = zy / x𝑋 = z / x𝑋)
6119, 60syl5eqr 2083 . . . . . . . . 9 (y = z𝑌 = z / x𝑋)
6261breq2d 3767 . . . . . . . 8 (y = z → (w / x𝑋𝑅𝑌w / x𝑋𝑅z / x𝑋))
6356, 57, 48, 58, 59, 62opelopabf 4002 . . . . . . 7 (⟨w, z {⟨x, y⟩ ∣ 𝑋𝑅𝑌} ↔ w / x𝑋𝑅z / x𝑋)
6454, 55, 633bitri 195 . . . . . 6 (w𝑆zw / x𝑋𝑅z / x𝑋)
6553, 64anbi12i 433 . . . . 5 ((v𝑆w w𝑆z) ↔ (v / x𝑋𝑅w / x𝑋 w / x𝑋𝑅z / x𝑋))
66 df-br 3756 . . . . . 6 (v𝑆z ↔ ⟨v, z 𝑆)
679eleq2i 2101 . . . . . 6 (⟨v, z 𝑆 ↔ ⟨v, z {⟨x, y⟩ ∣ 𝑋𝑅𝑌})
68 nfv 1418 . . . . . . 7 yv / x𝑋𝑅z / x𝑋
6961breq2d 3767 . . . . . . 7 (y = z → (v / x𝑋𝑅𝑌v / x𝑋𝑅z / x𝑋))
7013, 68, 15, 58, 16, 69opelopabf 4002 . . . . . 6 (⟨v, z {⟨x, y⟩ ∣ 𝑋𝑅𝑌} ↔ v / x𝑋𝑅z / x𝑋)
7166, 67, 703bitri 195 . . . . 5 (v𝑆zv / x𝑋𝑅z / x𝑋)
7244, 65, 713imtr4g 194 . . . 4 ((𝑅 Po B (v / x𝑋 B w / x𝑋 B z / x𝑋 B)) → ((v𝑆w w𝑆z) → v𝑆z))
7372adantlr 446 . . 3 (((𝑅 Po B x A 𝑋 B) (v / x𝑋 B w / x𝑋 B z / x𝑋 B)) → ((v𝑆w w𝑆z) → v𝑆z))
7443, 73syldan 266 . 2 (((𝑅 Po B x A 𝑋 B) (v A w A z A)) → ((v𝑆w w𝑆z) → v𝑆z))
7527, 74ispod 4032 1 ((𝑅 Po B x A 𝑋 B) → 𝑆 Po A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  wral 2300  csb 2846  cop 3370   class class class wbr 3755  {copab 3808   Po wpo 4022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-po 4024
This theorem is referenced by: (None)
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