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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
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