Theorem xpord2lem 8127

Description: Lemma for Cartesian product ordering. Calculate the value of the Cartesian product relation. (Contributed by Scott Fenton, 19-Aug-2024.)

Hypothesis

Ref	Expression
xpord2.1	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}

Assertion

Ref	Expression
xpord2lem	⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ ((𝑎𝑅𝑐 ∨ 𝑎 = 𝑐) ∧ (𝑏𝑆𝑑 ∨ 𝑏 = 𝑑) ∧ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑑))))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑎,𝑦   𝑥,𝑏,𝑦   𝑥,𝑐,𝑦   𝑥,𝑑,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐴(𝑎,𝑏,𝑐,𝑑)   𝐵(𝑎,𝑏,𝑐,𝑑)   𝑅(𝑎,𝑏,𝑐,𝑑)   𝑆(𝑎,𝑏,𝑐,𝑑)   𝑇(𝑥,𝑦,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem xpord2lem

Step	Hyp	Ref	Expression
1		opex 5464	. 2 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
2		opex 5464	. 2 ⊢ ⟨𝑐, 𝑑⟩ ∈ V
3		eleq1 2821	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ∈ (𝐴 × 𝐵) ↔ ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐵)))
4		opelxp 5712	. . . 4 ⊢ (⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))
5	3, 4	bitrdi 286	. . 3 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)))
6		vex 3478	. . . . . . 7 ⊢ 𝑎 ∈ V
7		vex 3478	. . . . . . 7 ⊢ 𝑏 ∈ V
8	6, 7	op1std 7984	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (1st ‘𝑥) = 𝑎)
9	8	breq1d 5158	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((1st ‘𝑥)𝑅(1st ‘𝑦) ↔ 𝑎𝑅(1st ‘𝑦)))
10	8	eqeq1d 2734	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((1st ‘𝑥) = (1st ‘𝑦) ↔ 𝑎 = (1st ‘𝑦)))
11	9, 10	orbi12d 917	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ↔ (𝑎𝑅(1st ‘𝑦) ∨ 𝑎 = (1st ‘𝑦))))
12	6, 7	op2ndd 7985	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (2nd ‘𝑥) = 𝑏)
13	12	breq1d 5158	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ↔ 𝑏𝑆(2nd ‘𝑦)))
14	12	eqeq1d 2734	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((2nd ‘𝑥) = (2nd ‘𝑦) ↔ 𝑏 = (2nd ‘𝑦)))
15	13, 14	orbi12d 917	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ↔ (𝑏𝑆(2nd ‘𝑦) ∨ 𝑏 = (2nd ‘𝑦))))
16		neeq1 3003	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ≠ 𝑦 ↔ ⟨𝑎, 𝑏⟩ ≠ 𝑦))
17	11, 15, 16	3anbi123d 1436	. . 3 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦) ↔ ((𝑎𝑅(1st ‘𝑦) ∨ 𝑎 = (1st ‘𝑦)) ∧ (𝑏𝑆(2nd ‘𝑦) ∨ 𝑏 = (2nd ‘𝑦)) ∧ ⟨𝑎, 𝑏⟩ ≠ 𝑦)))
18	5, 17	3anbi13d 1438	. 2 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ ((𝑎𝑅(1st ‘𝑦) ∨ 𝑎 = (1st ‘𝑦)) ∧ (𝑏𝑆(2nd ‘𝑦) ∨ 𝑏 = (2nd ‘𝑦)) ∧ ⟨𝑎, 𝑏⟩ ≠ 𝑦))))
19		eleq1 2821	. . . 4 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑦 ∈ (𝐴 × 𝐵) ↔ ⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵)))
20		opelxp 5712	. . . 4 ⊢ (⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵) ↔ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵))
21	19, 20	bitrdi 286	. . 3 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑦 ∈ (𝐴 × 𝐵) ↔ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)))
22		vex 3478	. . . . . . 7 ⊢ 𝑐 ∈ V
23		vex 3478	. . . . . . 7 ⊢ 𝑑 ∈ V
24	22, 23	op1std 7984	. . . . . 6 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (1st ‘𝑦) = 𝑐)
25	24	breq2d 5160	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑎𝑅(1st ‘𝑦) ↔ 𝑎𝑅𝑐))
26	24	eqeq2d 2743	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑎 = (1st ‘𝑦) ↔ 𝑎 = 𝑐))
27	25, 26	orbi12d 917	. . . 4 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → ((𝑎𝑅(1st ‘𝑦) ∨ 𝑎 = (1st ‘𝑦)) ↔ (𝑎𝑅𝑐 ∨ 𝑎 = 𝑐)))
28	22, 23	op2ndd 7985	. . . . . 6 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (2nd ‘𝑦) = 𝑑)
29	28	breq2d 5160	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑏𝑆(2nd ‘𝑦) ↔ 𝑏𝑆𝑑))
30	28	eqeq2d 2743	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑏 = (2nd ‘𝑦) ↔ 𝑏 = 𝑑))
31	29, 30	orbi12d 917	. . . 4 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → ((𝑏𝑆(2nd ‘𝑦) ∨ 𝑏 = (2nd ‘𝑦)) ↔ (𝑏𝑆𝑑 ∨ 𝑏 = 𝑑)))
32		neeq2 3004	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (⟨𝑎, 𝑏⟩ ≠ 𝑦 ↔ ⟨𝑎, 𝑏⟩ ≠ ⟨𝑐, 𝑑⟩))
33	6, 7	opthne 5482	. . . . 5 ⊢ (⟨𝑎, 𝑏⟩ ≠ ⟨𝑐, 𝑑⟩ ↔ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑑))
34	32, 33	bitrdi 286	. . . 4 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (⟨𝑎, 𝑏⟩ ≠ 𝑦 ↔ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑑)))
35	27, 31, 34	3anbi123d 1436	. . 3 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (((𝑎𝑅(1st ‘𝑦) ∨ 𝑎 = (1st ‘𝑦)) ∧ (𝑏𝑆(2nd ‘𝑦) ∨ 𝑏 = (2nd ‘𝑦)) ∧ ⟨𝑎, 𝑏⟩ ≠ 𝑦) ↔ ((𝑎𝑅𝑐 ∨ 𝑎 = 𝑐) ∧ (𝑏𝑆𝑑 ∨ 𝑏 = 𝑑) ∧ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑑))))
36	21, 35	3anbi23d 1439	. 2 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ ((𝑎𝑅(1st ‘𝑦) ∨ 𝑎 = (1st ‘𝑦)) ∧ (𝑏𝑆(2nd ‘𝑦) ∨ 𝑏 = (2nd ‘𝑦)) ∧ ⟨𝑎, 𝑏⟩ ≠ 𝑦)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ ((𝑎𝑅𝑐 ∨ 𝑎 = 𝑐) ∧ (𝑏𝑆𝑑 ∨ 𝑏 = 𝑑) ∧ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑑)))))
37		xpord2.1	. 2 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
38	1, 2, 18, 36, 37	brab 5543	1 ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ ((𝑎𝑅𝑐 ∨ 𝑎 = 𝑐) ∧ (𝑏𝑆𝑑 ∨ 𝑏 = 𝑑) ∧ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑑))))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ⟨cop 4634   class class class wbr 5148  {copab 5210   × cxp 5674  ‘cfv 6543  1^st c1st 7972  2^nd c2nd 7973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7974  df-2nd 7975

This theorem is referenced by:  poxp2  8128  frxp2  8129  xpord2pred  8130