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Theorem ltdfpr 6604
Description: More convenient form of df-iltp 6568. (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 3765 . . 3 (𝐴<P 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ <P )
2 df-iltp 6568 . . . 4 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))}
32eleq2i 2104 . . 3 (⟨𝐴, 𝐵⟩ ∈ <P ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))})
41, 3bitri 173 . 2 (𝐴<P 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))})
5 simpl 102 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
65fveq2d 5182 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (2nd𝑥) = (2nd𝐴))
76eleq2d 2107 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑞 ∈ (2nd𝑥) ↔ 𝑞 ∈ (2nd𝐴)))
8 simpr 103 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
98fveq2d 5182 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (1st𝑦) = (1st𝐵))
109eleq2d 2107 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑞 ∈ (1st𝑦) ↔ 𝑞 ∈ (1st𝐵)))
117, 10anbi12d 442 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)) ↔ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
1211rexbidv 2327 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
1312opelopab2a 4002 . 2 ((𝐴P𝐵P) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))} ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
144, 13syl5bb 181 1 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  wrex 2307  cop 3378   class class class wbr 3764  {copab 3817  cfv 4902  1st c1st 5765  2nd c2nd 5766  Qcnq 6378  Pcnp 6389  <P cltp 6393
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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-iota 4867  df-fv 4910  df-iltp 6568
This theorem is referenced by:  nqprl  6649  nqpru  6650  ltprordil  6687  ltnqpr  6691  ltnqpri  6692  ltpopr  6693  ltsopr  6694  ltaddpr  6695  ltexprlemm  6698  ltexprlemopu  6701  ltexprlemru  6710  aptiprleml  6737  aptiprlemu  6738  archpr  6741  cauappcvgprlem2  6758  caucvgprlem2  6778  caucvgprprlemopu  6797  caucvgprprlemexbt  6804  caucvgprprlem2  6808
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