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Theorem xporderlem 7448
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 4797 . . 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 2823 . . 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 5073 . . 3 𝑎, 𝑏⟩ ∈ V
6 opex 5073 . . 3 𝑐, 𝑑⟩ ∈ V
7 eleq1 2819 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ∈ (𝐴 × 𝐵) ↔ ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐵)))
8 opelxp 5295 . . . . . 6 (⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐵) ↔ (𝑎𝐴𝑏𝐵))
97, 8syl6bb 276 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ∈ (𝐴 × 𝐵) ↔ (𝑎𝐴𝑏𝐵)))
109anbi1d 743 . . . 4 (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ↔ ((𝑎𝐴𝑏𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵))))
11 vex 3335 . . . . . . 7 𝑎 ∈ V
12 vex 3335 . . . . . . 7 𝑏 ∈ V
1311, 12op1std 7335 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏⟩ → (1st𝑥) = 𝑎)
1413breq1d 4806 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → ((1st𝑥)𝑅(1st𝑦) ↔ 𝑎𝑅(1st𝑦)))
1513eqeq1d 2754 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏⟩ → ((1st𝑥) = (1st𝑦) ↔ 𝑎 = (1st𝑦)))
1611, 12op2ndd 7336 . . . . . . 7 (𝑥 = ⟨𝑎, 𝑏⟩ → (2nd𝑥) = 𝑏)
1716breq1d 4806 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏⟩ → ((2nd𝑥)𝑆(2nd𝑦) ↔ 𝑏𝑆(2nd𝑦)))
1815, 17anbi12d 749 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦)) ↔ (𝑎 = (1st𝑦) ∧ 𝑏𝑆(2nd𝑦))))
1914, 18orbi12d 748 . . . 4 (𝑥 = ⟨𝑎, 𝑏⟩ → (((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))) ↔ (𝑎𝑅(1st𝑦) ∨ (𝑎 = (1st𝑦) ∧ 𝑏𝑆(2nd𝑦)))))
2010, 19anbi12d 749 . . 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 2819 . . . . . 6 (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑦 ∈ (𝐴 × 𝐵) ↔ ⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵)))
22 opelxp 5295 . . . . . 6 (⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵) ↔ (𝑐𝐴𝑑𝐵))
2321, 22syl6bb 276 . . . . 5 (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑦 ∈ (𝐴 × 𝐵) ↔ (𝑐𝐴𝑑𝐵)))
2423anbi2d 742 . . . 4 (𝑦 = ⟨𝑐, 𝑑⟩ → (((𝑎𝐴𝑏𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝑐𝐴𝑑𝐵))))
25 vex 3335 . . . . . . 7 𝑐 ∈ V
26 vex 3335 . . . . . . 7 𝑑 ∈ V
2725, 26op1std 7335 . . . . . 6 (𝑦 = ⟨𝑐, 𝑑⟩ → (1st𝑦) = 𝑐)
2827breq2d 4808 . . . . 5 (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑎𝑅(1st𝑦) ↔ 𝑎𝑅𝑐))
2927eqeq2d 2762 . . . . . 6 (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑎 = (1st𝑦) ↔ 𝑎 = 𝑐))
3025, 26op2ndd 7336 . . . . . . 7 (𝑦 = ⟨𝑐, 𝑑⟩ → (2nd𝑦) = 𝑑)
3130breq2d 4808 . . . . . 6 (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑏𝑆(2nd𝑦) ↔ 𝑏𝑆𝑑))
3229, 31anbi12d 749 . . . . 5 (𝑦 = ⟨𝑐, 𝑑⟩ → ((𝑎 = (1st𝑦) ∧ 𝑏𝑆(2nd𝑦)) ↔ (𝑎 = 𝑐𝑏𝑆𝑑)))
3328, 32orbi12d 748 . . . 4 (𝑦 = ⟨𝑐, 𝑑⟩ → ((𝑎𝑅(1st𝑦) ∨ (𝑎 = (1st𝑦) ∧ 𝑏𝑆(2nd𝑦))) ↔ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐𝑏𝑆𝑑))))
3424, 33anbi12d 749 . . 3 (𝑦 = ⟨𝑐, 𝑑⟩ → ((((𝑎𝐴𝑏𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ (𝑎𝑅(1st𝑦) ∨ (𝑎 = (1st𝑦) ∧ 𝑏𝑆(2nd𝑦)))) ↔ (((𝑎𝐴𝑏𝐵) ∧ (𝑐𝐴𝑑𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐𝑏𝑆𝑑)))))
355, 6, 20, 34opelopab 5139 . 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 900 . . 3 (((𝑎𝐴𝑏𝐵) ∧ (𝑐𝐴𝑑𝐵)) ↔ ((𝑎𝐴𝑐𝐴) ∧ (𝑏𝐵𝑑𝐵)))
3736anbi1i 733 . 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 382  wa 383   = wceq 1624  wcel 2131  cop 4319   class class class wbr 4796  {copab 4856   × cxp 5256  cfv 6041  1st c1st 7323  2nd c2nd 7324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fv 6049  df-1st 7325  df-2nd 7326
This theorem is referenced by:  poxp  7449  soxp  7450
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