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Theorem ltdfpr 7337
Description: More convenient form of df-iltp 7301. (Contributed by Jim Kingdon, 15-Dec-2019.)
Assertion
Ref Expression
ltdfpr ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
Distinct variable groups:   𝐴,𝑞   𝐵,𝑞

Proof of Theorem ltdfpr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 3937 . . 3 (𝐴<P 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ <P )
2 df-iltp 7301 . . . 4 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))}
32eleq2i 2207 . . 3 (⟨𝐴, 𝐵⟩ ∈ <P ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))})
41, 3bitri 183 . 2 (𝐴<P 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))})
5 simpl 108 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
65fveq2d 5432 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (2nd𝑥) = (2nd𝐴))
76eleq2d 2210 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑞 ∈ (2nd𝑥) ↔ 𝑞 ∈ (2nd𝐴)))
8 simpr 109 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
98fveq2d 5432 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (1st𝑦) = (1st𝐵))
109eleq2d 2210 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑞 ∈ (1st𝑦) ↔ 𝑞 ∈ (1st𝐵)))
117, 10anbi12d 465 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)) ↔ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
1211rexbidv 2439 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
1312opelopab2a 4194 . 2 ((𝐴P𝐵P) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))} ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
144, 13syl5bb 191 1 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  wrex 2418  cop 3534   class class class wbr 3936  {copab 3995  cfv 5130  1st c1st 6043  2nd c2nd 6044  Qcnq 7111  Pcnp 7122  <P cltp 7126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-iota 5095  df-fv 5138  df-iltp 7301
This theorem is referenced by:  nqprl  7382  nqpru  7383  ltprordil  7420  ltnqpr  7424  ltnqpri  7425  ltpopr  7426  ltsopr  7427  ltaddpr  7428  ltexprlemm  7431  ltexprlemopu  7434  ltexprlemru  7443  aptiprleml  7470  aptiprlemu  7471  archpr  7474  cauappcvgprlem2  7491  caucvgprlem2  7511  caucvgprprlemopu  7530  caucvgprprlemexbt  7537  caucvgprprlem2  7541  suplocexprlemloc  7552  suplocexprlemub  7554  suplocexprlemlub  7555
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