Theorem ltdfpr 6489

Description: More convenient form of df-iltp 6453. (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 3756	. . 3 ⊢ (A<P B ↔ ⟨A, B⟩ ∈ <P )
2		df-iltp 6453	. . . 4 ⊢ <P = {⟨x, y⟩ ∣ ((x ∈ P ∧ y ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)))}
3	2	eleq2i 2101	. . 3 ⊢ (⟨A, B⟩ ∈ <P ↔ ⟨A, B⟩ ∈ {⟨x, y⟩ ∣ ((x ∈ P ∧ y ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)))})
4	1, 3	bitri 173	. 2 ⊢ (A<P B ↔ ⟨A, B⟩ ∈ {⟨x, y⟩ ∣ ((x ∈ P ∧ y ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)))})
5		simpl 102	. . . . . . 7 ⊢ ((x = A ∧ y = B) → x = A)
6	5	fveq2d 5125	. . . . . 6 ⊢ ((x = A ∧ y = B) → (2nd ‘x) = (2nd ‘A))
7	6	eleq2d 2104	. . . . 5 ⊢ ((x = A ∧ y = B) → (𝑞 ∈ (2nd ‘x) ↔ 𝑞 ∈ (2nd ‘A)))
8		simpr 103	. . . . . . 7 ⊢ ((x = A ∧ y = B) → y = B)
9	8	fveq2d 5125	. . . . . 6 ⊢ ((x = A ∧ y = B) → (1st ‘y) = (1st ‘B))
10	9	eleq2d 2104	. . . . 5 ⊢ ((x = A ∧ y = B) → (𝑞 ∈ (1st ‘y) ↔ 𝑞 ∈ (1st ‘B)))
11	7, 10	anbi12d 442	. . . 4 ⊢ ((x = A ∧ y = B) → ((𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)) ↔ (𝑞 ∈ (2nd ‘A) ∧ 𝑞 ∈ (1st ‘B))))
12	11	rexbidv 2321	. . 3 ⊢ ((x = A ∧ y = B) → (∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)) ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘A) ∧ 𝑞 ∈ (1st ‘B))))
13	12	opelopab2a 3993	. 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	syl5bb 181	1 ⊢ ((A ∈ P ∧ B ∈ P) → (A<P B ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘A) ∧ 𝑞 ∈ (1st ‘B))))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  ⟨cop 3370   class class class wbr 3755  {copab 3808  ‘cfv 4845  1^st c1st 5707  2^nd c2nd 5708  Qcnq 6264  Pcnp 6275  <_P cltp 6279

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-iota 4810  df-fv 4853  df-iltp 6453

This theorem is referenced by:  nqprl  6533  ltprordil  6565  ltpopr  6569  ltsopr  6570  ltaddpr  6571  ltexprlemm  6574  ltexprlemopu  6577  ltexprlemru  6586  aptiprleml  6611  aptiprlemu  6612  archpr  6615  cauappcvgprlem2  6632  caucvgprlem2  6651