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Theorem xporderlem 7233
Description: Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
Hypothesis
Ref Expression
xporderlem.1 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))}
Assertion
Ref Expression
xporderlem (⟨𝑎, 𝑏𝑇𝑐, 𝑑⟩ ↔ (((𝑎𝐴𝑐𝐴) ∧ (𝑏𝐵𝑑𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐𝑏𝑆𝑑))))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑎,𝑦   𝑥,𝑏,𝑦   𝑥,𝑐,𝑦   𝑥,𝑑,𝑦
Allowed substitution hints:   𝐴(𝑎,𝑏,𝑐,𝑑)   𝐵(𝑎,𝑏,𝑐,𝑑)   𝑅(𝑎,𝑏,𝑐,𝑑)   𝑆(𝑎,𝑏,𝑐,𝑑)   𝑇(𝑥,𝑦,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem xporderlem
StepHypRef Expression
1 df-br 4614 . . 3 (⟨𝑎, 𝑏𝑇𝑐, 𝑑⟩ ↔ ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ 𝑇)
2 xporderlem.1 . . . 4 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))}
32eleq2i 2690 . . 3 (⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ 𝑇 ↔ ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))})
41, 3bitri 264 . 2 (⟨𝑎, 𝑏𝑇𝑐, 𝑑⟩ ↔ ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))})
5 opex 4893 . . 3 𝑎, 𝑏⟩ ∈ V
6 opex 4893 . . 3 𝑐, 𝑑⟩ ∈ V
7 eleq1 2686 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ∈ (𝐴 × 𝐵) ↔ ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐵)))
8 opelxp 5106 . . . . . 6 (⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐵) ↔ (𝑎𝐴𝑏𝐵))
97, 8syl6bb 276 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ∈ (𝐴 × 𝐵) ↔ (𝑎𝐴𝑏𝐵)))
109anbi1d 740 . . . 4 (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ↔ ((𝑎𝐴𝑏𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵))))
11 vex 3189 . . . . . . 7 𝑎 ∈ V
12 vex 3189 . . . . . . 7 𝑏 ∈ V
1311, 12op1std 7123 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏⟩ → (1st𝑥) = 𝑎)
1413breq1d 4623 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → ((1st𝑥)𝑅(1st𝑦) ↔ 𝑎𝑅(1st𝑦)))
1513eqeq1d 2623 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏⟩ → ((1st𝑥) = (1st𝑦) ↔ 𝑎 = (1st𝑦)))
1611, 12op2ndd 7124 . . . . . . 7 (𝑥 = ⟨𝑎, 𝑏⟩ → (2nd𝑥) = 𝑏)
1716breq1d 4623 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏⟩ → ((2nd𝑥)𝑆(2nd𝑦) ↔ 𝑏𝑆(2nd𝑦)))
1815, 17anbi12d 746 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦)) ↔ (𝑎 = (1st𝑦) ∧ 𝑏𝑆(2nd𝑦))))
1914, 18orbi12d 745 . . . 4 (𝑥 = ⟨𝑎, 𝑏⟩ → (((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))) ↔ (𝑎𝑅(1st𝑦) ∨ (𝑎 = (1st𝑦) ∧ 𝑏𝑆(2nd𝑦)))))
2010, 19anbi12d 746 . . 3 (𝑥 = ⟨𝑎, 𝑏⟩ → (((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦)))) ↔ (((𝑎𝐴𝑏𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ (𝑎𝑅(1st𝑦) ∨ (𝑎 = (1st𝑦) ∧ 𝑏𝑆(2nd𝑦))))))
21 eleq1 2686 . . . . . 6 (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑦 ∈ (𝐴 × 𝐵) ↔ ⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵)))
22 opelxp 5106 . . . . . 6 (⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵) ↔ (𝑐𝐴𝑑𝐵))
2321, 22syl6bb 276 . . . . 5 (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑦 ∈ (𝐴 × 𝐵) ↔ (𝑐𝐴𝑑𝐵)))
2423anbi2d 739 . . . 4 (𝑦 = ⟨𝑐, 𝑑⟩ → (((𝑎𝐴𝑏𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝑐𝐴𝑑𝐵))))
25 vex 3189 . . . . . . 7 𝑐 ∈ V
26 vex 3189 . . . . . . 7 𝑑 ∈ V
2725, 26op1std 7123 . . . . . 6 (𝑦 = ⟨𝑐, 𝑑⟩ → (1st𝑦) = 𝑐)
2827breq2d 4625 . . . . 5 (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑎𝑅(1st𝑦) ↔ 𝑎𝑅𝑐))
2927eqeq2d 2631 . . . . . 6 (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑎 = (1st𝑦) ↔ 𝑎 = 𝑐))
3025, 26op2ndd 7124 . . . . . . 7 (𝑦 = ⟨𝑐, 𝑑⟩ → (2nd𝑦) = 𝑑)
3130breq2d 4625 . . . . . 6 (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑏𝑆(2nd𝑦) ↔ 𝑏𝑆𝑑))
3229, 31anbi12d 746 . . . . 5 (𝑦 = ⟨𝑐, 𝑑⟩ → ((𝑎 = (1st𝑦) ∧ 𝑏𝑆(2nd𝑦)) ↔ (𝑎 = 𝑐𝑏𝑆𝑑)))
3328, 32orbi12d 745 . . . 4 (𝑦 = ⟨𝑐, 𝑑⟩ → ((𝑎𝑅(1st𝑦) ∨ (𝑎 = (1st𝑦) ∧ 𝑏𝑆(2nd𝑦))) ↔ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐𝑏𝑆𝑑))))
3424, 33anbi12d 746 . . 3 (𝑦 = ⟨𝑐, 𝑑⟩ → ((((𝑎𝐴𝑏𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ (𝑎𝑅(1st𝑦) ∨ (𝑎 = (1st𝑦) ∧ 𝑏𝑆(2nd𝑦)))) ↔ (((𝑎𝐴𝑏𝐵) ∧ (𝑐𝐴𝑑𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐𝑏𝑆𝑑)))))
355, 6, 20, 34opelopab 4957 . 2 (⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))} ↔ (((𝑎𝐴𝑏𝐵) ∧ (𝑐𝐴𝑑𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐𝑏𝑆𝑑))))
36 an4 864 . . 3 (((𝑎𝐴𝑏𝐵) ∧ (𝑐𝐴𝑑𝐵)) ↔ ((𝑎𝐴𝑐𝐴) ∧ (𝑏𝐵𝑑𝐵)))
3736anbi1i 730 . 2 ((((𝑎𝐴𝑏𝐵) ∧ (𝑐𝐴𝑑𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐𝑏𝑆𝑑))) ↔ (((𝑎𝐴𝑐𝐴) ∧ (𝑏𝐵𝑑𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐𝑏𝑆𝑑))))
384, 35, 373bitri 286 1 (⟨𝑎, 𝑏𝑇𝑐, 𝑑⟩ ↔ (((𝑎𝐴𝑐𝐴) ∧ (𝑏𝐵𝑑𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐𝑏𝑆𝑑))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  cop 4154   class class class wbr 4613  {copab 4672   × cxp 5072  cfv 5847  1st c1st 7111  2nd c2nd 7112
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-1st 7113  df-2nd 7114
This theorem is referenced by:  poxp  7234  soxp  7235
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