Theorem ltdfpr 7704

Description: More convenient form of df-iltp 7668. (Contributed by Jim Kingdon, 15-Dec-2019.)

Assertion

Ref	Expression
ltdfpr	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵))))

Distinct variable groups:   𝐴,𝑞   𝐵,𝑞

Proof of Theorem ltdfpr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 4084	. . 3 ⊢ (𝐴<P 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ <P )
2		df-iltp 7668	. . . 4 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
3	2	eleq2i 2296	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ <P ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))})
4	1, 3	bitri 184	. 2 ⊢ (𝐴<P 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))})
5		simpl 109	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
6	5	fveq2d 5633	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (2nd ‘𝑥) = (2nd ‘𝐴))
7	6	eleq2d 2299	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑞 ∈ (2nd ‘𝑥) ↔ 𝑞 ∈ (2nd ‘𝐴)))
8		simpr 110	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
9	8	fveq2d 5633	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (1st ‘𝑦) = (1st ‘𝐵))
10	9	eleq2d 2299	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑞 ∈ (1st ‘𝑦) ↔ 𝑞 ∈ (1st ‘𝐵)))
11	7, 10	anbi12d 473	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)) ↔ (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵))))
12	11	rexbidv 2531	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)) ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵))))
13	12	opelopab2a 4353	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵))))
14	4, 13	bitrid 192	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  ⟨cop 3669   class class class wbr 4083  {copab 4144  ‘cfv 5318  1^st c1st 6290  2^nd c2nd 6291  Qcnq 7478  Pcnp 7489  <_P cltp 7493

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-iota 5278  df-fv 5326  df-iltp 7668

This theorem is referenced by:  nqprl  7749  nqpru  7750  ltprordil  7787  ltnqpr  7791  ltnqpri  7792  ltpopr  7793  ltsopr  7794  ltaddpr  7795  ltexprlemm  7798  ltexprlemopu  7801  ltexprlemru  7810  aptiprleml  7837  aptiprlemu  7838  archpr  7841  cauappcvgprlem2  7858  caucvgprlem2  7878  caucvgprprlemopu  7897  caucvgprprlemexbt  7904  caucvgprprlem2  7908  suplocexprlemloc  7919  suplocexprlemub  7921  suplocexprlemlub  7922