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Theorem ltdfpr 7493
Description: More convenient form of df-iltp 7457. (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 4001 . . 3 (𝐴<P 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ <P )
2 df-iltp 7457 . . . 4 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))}
32eleq2i 2244 . . 3 (⟨𝐴, 𝐵⟩ ∈ <P ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))})
41, 3bitri 184 . 2 (𝐴<P 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))})
5 simpl 109 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
65fveq2d 5515 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (2nd𝑥) = (2nd𝐴))
76eleq2d 2247 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑞 ∈ (2nd𝑥) ↔ 𝑞 ∈ (2nd𝐴)))
8 simpr 110 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
98fveq2d 5515 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (1st𝑦) = (1st𝐵))
109eleq2d 2247 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑞 ∈ (1st𝑦) ↔ 𝑞 ∈ (1st𝐵)))
117, 10anbi12d 473 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)) ↔ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
1211rexbidv 2478 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
1312opelopab2a 4262 . 2 ((𝐴P𝐵P) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))} ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
144, 13bitrid 192 1 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wrex 2456  cop 3594   class class class wbr 4000  {copab 4060  cfv 5212  1st c1st 6133  2nd c2nd 6134  Qcnq 7267  Pcnp 7278  <P cltp 7282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-iota 5174  df-fv 5220  df-iltp 7457
This theorem is referenced by:  nqprl  7538  nqpru  7539  ltprordil  7576  ltnqpr  7580  ltnqpri  7581  ltpopr  7582  ltsopr  7583  ltaddpr  7584  ltexprlemm  7587  ltexprlemopu  7590  ltexprlemru  7599  aptiprleml  7626  aptiprlemu  7627  archpr  7630  cauappcvgprlem2  7647  caucvgprlem2  7667  caucvgprprlemopu  7686  caucvgprprlemexbt  7693  caucvgprprlem2  7697  suplocexprlemloc  7708  suplocexprlemub  7710  suplocexprlemlub  7711
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