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Theorem pofun 5016
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 3534 . . . . . . 7 𝑥𝑣 / 𝑥𝑋
21nfel1 2775 . . . . . 6 𝑥𝑣 / 𝑥𝑋𝐵
3 csbeq1a 3527 . . . . . . 7 (𝑥 = 𝑣𝑋 = 𝑣 / 𝑥𝑋)
43eleq1d 2683 . . . . . 6 (𝑥 = 𝑣 → (𝑋𝐵𝑣 / 𝑥𝑋𝐵))
52, 4rspc 3292 . . . . 5 (𝑣𝐴 → (∀𝑥𝐴 𝑋𝐵𝑣 / 𝑥𝑋𝐵))
65impcom 446 . . . 4 ((∀𝑥𝐴 𝑋𝐵𝑣𝐴) → 𝑣 / 𝑥𝑋𝐵)
7 poirr 5011 . . . . 5 ((𝑅 Po 𝐵𝑣 / 𝑥𝑋𝐵) → ¬ 𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋)
8 df-br 4619 . . . . . 6 (𝑣𝑆𝑣 ↔ ⟨𝑣, 𝑣⟩ ∈ 𝑆)
9 pofun.1 . . . . . . 7 𝑆 = {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌}
109eleq2i 2690 . . . . . 6 (⟨𝑣, 𝑣⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
11 nfcv 2761 . . . . . . . 8 𝑥𝑅
12 nfcv 2761 . . . . . . . 8 𝑥𝑌
131, 11, 12nfbr 4664 . . . . . . 7 𝑥𝑣 / 𝑥𝑋𝑅𝑌
14 nfv 1840 . . . . . . 7 𝑦𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋
15 vex 3192 . . . . . . 7 𝑣 ∈ V
163breq1d 4628 . . . . . . 7 (𝑥 = 𝑣 → (𝑋𝑅𝑌𝑣 / 𝑥𝑋𝑅𝑌))
17 vex 3192 . . . . . . . . . 10 𝑦 ∈ V
18 pofun.2 . . . . . . . . . 10 (𝑥 = 𝑦𝑋 = 𝑌)
1917, 18csbie 3544 . . . . . . . . 9 𝑦 / 𝑥𝑋 = 𝑌
20 csbeq1 3521 . . . . . . . . 9 (𝑦 = 𝑣𝑦 / 𝑥𝑋 = 𝑣 / 𝑥𝑋)
2119, 20syl5eqr 2669 . . . . . . . 8 (𝑦 = 𝑣𝑌 = 𝑣 / 𝑥𝑋)
2221breq2d 4630 . . . . . . 7 (𝑦 = 𝑣 → (𝑣 / 𝑥𝑋𝑅𝑌𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋))
2313, 14, 15, 15, 16, 22opelopabf 4965 . . . . . 6 (⟨𝑣, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ 𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋)
248, 10, 233bitri 286 . . . . 5 (𝑣𝑆𝑣𝑣 / 𝑥𝑋𝑅𝑣 / 𝑥𝑋)
257, 24sylnibr 319 . . . 4 ((𝑅 Po 𝐵𝑣 / 𝑥𝑋𝐵) → ¬ 𝑣𝑆𝑣)
266, 25sylan2 491 . . 3 ((𝑅 Po 𝐵 ∧ (∀𝑥𝐴 𝑋𝐵𝑣𝐴)) → ¬ 𝑣𝑆𝑣)
2726anassrs 679 . 2 (((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) ∧ 𝑣𝐴) → ¬ 𝑣𝑆𝑣)
285com12 32 . . . . . 6 (∀𝑥𝐴 𝑋𝐵 → (𝑣𝐴𝑣 / 𝑥𝑋𝐵))
29 nfcsb1v 3534 . . . . . . . . 9 𝑥𝑤 / 𝑥𝑋
3029nfel1 2775 . . . . . . . 8 𝑥𝑤 / 𝑥𝑋𝐵
31 csbeq1a 3527 . . . . . . . . 9 (𝑥 = 𝑤𝑋 = 𝑤 / 𝑥𝑋)
3231eleq1d 2683 . . . . . . . 8 (𝑥 = 𝑤 → (𝑋𝐵𝑤 / 𝑥𝑋𝐵))
3330, 32rspc 3292 . . . . . . 7 (𝑤𝐴 → (∀𝑥𝐴 𝑋𝐵𝑤 / 𝑥𝑋𝐵))
3433com12 32 . . . . . 6 (∀𝑥𝐴 𝑋𝐵 → (𝑤𝐴𝑤 / 𝑥𝑋𝐵))
35 nfcsb1v 3534 . . . . . . . . 9 𝑥𝑧 / 𝑥𝑋
3635nfel1 2775 . . . . . . . 8 𝑥𝑧 / 𝑥𝑋𝐵
37 csbeq1a 3527 . . . . . . . . 9 (𝑥 = 𝑧𝑋 = 𝑧 / 𝑥𝑋)
3837eleq1d 2683 . . . . . . . 8 (𝑥 = 𝑧 → (𝑋𝐵𝑧 / 𝑥𝑋𝐵))
3936, 38rspc 3292 . . . . . . 7 (𝑧𝐴 → (∀𝑥𝐴 𝑋𝐵𝑧 / 𝑥𝑋𝐵))
4039com12 32 . . . . . 6 (∀𝑥𝐴 𝑋𝐵 → (𝑧𝐴𝑧 / 𝑥𝑋𝐵))
4128, 34, 403anim123d 1403 . . . . 5 (∀𝑥𝐴 𝑋𝐵 → ((𝑣𝐴𝑤𝐴𝑧𝐴) → (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵)))
4241imp 445 . . . 4 ((∀𝑥𝐴 𝑋𝐵 ∧ (𝑣𝐴𝑤𝐴𝑧𝐴)) → (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵))
4342adantll 749 . . 3 (((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) ∧ (𝑣𝐴𝑤𝐴𝑧𝐴)) → (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵))
44 potr 5012 . . . . 5 ((𝑅 Po 𝐵 ∧ (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵)) → ((𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋) → 𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋))
45 df-br 4619 . . . . . . 7 (𝑣𝑆𝑤 ↔ ⟨𝑣, 𝑤⟩ ∈ 𝑆)
469eleq2i 2690 . . . . . . 7 (⟨𝑣, 𝑤⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
47 nfv 1840 . . . . . . . 8 𝑦𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋
48 vex 3192 . . . . . . . 8 𝑤 ∈ V
49 csbeq1 3521 . . . . . . . . . 10 (𝑦 = 𝑤𝑦 / 𝑥𝑋 = 𝑤 / 𝑥𝑋)
5019, 49syl5eqr 2669 . . . . . . . . 9 (𝑦 = 𝑤𝑌 = 𝑤 / 𝑥𝑋)
5150breq2d 4630 . . . . . . . 8 (𝑦 = 𝑤 → (𝑣 / 𝑥𝑋𝑅𝑌𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋))
5213, 47, 15, 48, 16, 51opelopabf 4965 . . . . . . 7 (⟨𝑣, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ 𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋)
5345, 46, 523bitri 286 . . . . . 6 (𝑣𝑆𝑤𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋)
54 df-br 4619 . . . . . . 7 (𝑤𝑆𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑆)
559eleq2i 2690 . . . . . . 7 (⟨𝑤, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
5629, 11, 12nfbr 4664 . . . . . . . 8 𝑥𝑤 / 𝑥𝑋𝑅𝑌
57 nfv 1840 . . . . . . . 8 𝑦𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋
58 vex 3192 . . . . . . . 8 𝑧 ∈ V
5931breq1d 4628 . . . . . . . 8 (𝑥 = 𝑤 → (𝑋𝑅𝑌𝑤 / 𝑥𝑋𝑅𝑌))
60 csbeq1 3521 . . . . . . . . . 10 (𝑦 = 𝑧𝑦 / 𝑥𝑋 = 𝑧 / 𝑥𝑋)
6119, 60syl5eqr 2669 . . . . . . . . 9 (𝑦 = 𝑧𝑌 = 𝑧 / 𝑥𝑋)
6261breq2d 4630 . . . . . . . 8 (𝑦 = 𝑧 → (𝑤 / 𝑥𝑋𝑅𝑌𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋))
6356, 57, 48, 58, 59, 62opelopabf 4965 . . . . . . 7 (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ 𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋)
6454, 55, 633bitri 286 . . . . . 6 (𝑤𝑆𝑧𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋)
6553, 64anbi12i 732 . . . . 5 ((𝑣𝑆𝑤𝑤𝑆𝑧) ↔ (𝑣 / 𝑥𝑋𝑅𝑤 / 𝑥𝑋𝑤 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋))
66 df-br 4619 . . . . . 6 (𝑣𝑆𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ 𝑆)
679eleq2i 2690 . . . . . 6 (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌})
68 nfv 1840 . . . . . . 7 𝑦𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋
6961breq2d 4630 . . . . . . 7 (𝑦 = 𝑧 → (𝑣 / 𝑥𝑋𝑅𝑌𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋))
7013, 68, 15, 58, 16, 69opelopabf 4965 . . . . . 6 (⟨𝑣, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} ↔ 𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋)
7166, 67, 703bitri 286 . . . . 5 (𝑣𝑆𝑧𝑣 / 𝑥𝑋𝑅𝑧 / 𝑥𝑋)
7244, 65, 713imtr4g 285 . . . 4 ((𝑅 Po 𝐵 ∧ (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵)) → ((𝑣𝑆𝑤𝑤𝑆𝑧) → 𝑣𝑆𝑧))
7372adantlr 750 . . 3 (((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) ∧ (𝑣 / 𝑥𝑋𝐵𝑤 / 𝑥𝑋𝐵𝑧 / 𝑥𝑋𝐵)) → ((𝑣𝑆𝑤𝑤𝑆𝑧) → 𝑣𝑆𝑧))
7443, 73syldan 487 . 2 (((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) ∧ (𝑣𝐴𝑤𝐴𝑧𝐴)) → ((𝑣𝑆𝑤𝑤𝑆𝑧) → 𝑣𝑆𝑧))
7527, 74ispod 5008 1 ((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) → 𝑆 Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  csb 3518  cop 4159   class class class wbr 4618  {copab 4677   Po wpo 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-po 5000
This theorem is referenced by: (None)
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