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Theorem ltdfpr 6662
Description: More convenient form of df-iltp 6626. (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 3793 . . 3 (𝐴<P 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ <P )
2 df-iltp 6626 . . . 4 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))}
32eleq2i 2120 . . 3 (⟨𝐴, 𝐵⟩ ∈ <P ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))})
41, 3bitri 177 . 2 (𝐴<P 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))})
5 simpl 106 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
65fveq2d 5210 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (2nd𝑥) = (2nd𝐴))
76eleq2d 2123 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑞 ∈ (2nd𝑥) ↔ 𝑞 ∈ (2nd𝐴)))
8 simpr 107 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
98fveq2d 5210 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (1st𝑦) = (1st𝐵))
109eleq2d 2123 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑞 ∈ (1st𝑦) ↔ 𝑞 ∈ (1st𝐵)))
117, 10anbi12d 450 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)) ↔ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
1211rexbidv 2344 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
1312opelopab2a 4030 . 2 ((𝐴P𝐵P) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))} ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
144, 13syl5bb 185 1 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wrex 2324  cop 3406   class class class wbr 3792  {copab 3845  cfv 4930  1st c1st 5793  2nd c2nd 5794  Qcnq 6436  Pcnp 6447  <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-iota 4895  df-fv 4938  df-iltp 6626
This theorem is referenced by:  nqprl  6707  nqpru  6708  ltprordil  6745  ltnqpr  6749  ltnqpri  6750  ltpopr  6751  ltsopr  6752  ltaddpr  6753  ltexprlemm  6756  ltexprlemopu  6759  ltexprlemru  6768  aptiprleml  6795  aptiprlemu  6796  archpr  6799  cauappcvgprlem2  6816  caucvgprlem2  6836  caucvgprprlemopu  6855  caucvgprprlemexbt  6862  caucvgprprlem2  6866
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