Theorem xpord2lem 8137

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 5446	. 2 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
2		opex 5446	. 2 ⊢ ⟨𝑐, 𝑑⟩ ∈ V
3		eleq1 2857	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ∈ (𝐴 × 𝐵) ↔ ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐵)))
4		opelxp 5698	. . . 4 ⊢ (⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))
5	3, 4	bitrdi 290	. . 3 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)))
6		vex 3467	. . . . . . 7 ⊢ 𝑎 ∈ V
7		vex 3467	. . . . . . 7 ⊢ 𝑏 ∈ V
8	6, 7	op1std 7995	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (1st ‘𝑥) = 𝑎)
9	8	breq1d 5123	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((1st ‘𝑥)𝑅(1st ‘𝑦) ↔ 𝑎𝑅(1st ‘𝑦)))
10	8	eqeq1d 2771	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((1st ‘𝑥) = (1st ‘𝑦) ↔ 𝑎 = (1st ‘𝑦)))
11	9, 10	orbi12d 931	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ↔ (𝑎𝑅(1st ‘𝑦) ∨ 𝑎 = (1st ‘𝑦))))
12	6, 7	op2ndd 7996	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (2nd ‘𝑥) = 𝑏)
13	12	breq1d 5123	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ↔ 𝑏𝑆(2nd ‘𝑦)))
14	12	eqeq1d 2771	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((2nd ‘𝑥) = (2nd ‘𝑦) ↔ 𝑏 = (2nd ‘𝑦)))
15	13, 14	orbi12d 931	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ↔ (𝑏𝑆(2nd ‘𝑦) ∨ 𝑏 = (2nd ‘𝑦))))
16		neeq1 3026	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ≠ 𝑦 ↔ ⟨𝑎, 𝑏⟩ ≠ 𝑦))
17	11, 15, 16	3anbi123d 1462	. . 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 1464	. 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 2857	. . . 4 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑦 ∈ (𝐴 × 𝐵) ↔ ⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵)))
20		opelxp 5698	. . . 4 ⊢ (⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵) ↔ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵))
21	19, 20	bitrdi 290	. . 3 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑦 ∈ (𝐴 × 𝐵) ↔ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)))
22		vex 3467	. . . . . . 7 ⊢ 𝑐 ∈ V
23		vex 3467	. . . . . . 7 ⊢ 𝑑 ∈ V
24	22, 23	op1std 7995	. . . . . 6 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (1st ‘𝑦) = 𝑐)
25	24	breq2d 5125	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑎𝑅(1st ‘𝑦) ↔ 𝑎𝑅𝑐))
26	24	eqeq2d 2780	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑎 = (1st ‘𝑦) ↔ 𝑎 = 𝑐))
27	25, 26	orbi12d 931	. . . 4 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → ((𝑎𝑅(1st ‘𝑦) ∨ 𝑎 = (1st ‘𝑦)) ↔ (𝑎𝑅𝑐 ∨ 𝑎 = 𝑐)))
28	22, 23	op2ndd 7996	. . . . . 6 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (2nd ‘𝑦) = 𝑑)
29	28	breq2d 5125	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑏𝑆(2nd ‘𝑦) ↔ 𝑏𝑆𝑑))
30	28	eqeq2d 2780	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑏 = (2nd ‘𝑦) ↔ 𝑏 = 𝑑))
31	29, 30	orbi12d 931	. . . 4 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → ((𝑏𝑆(2nd ‘𝑦) ∨ 𝑏 = (2nd ‘𝑦)) ↔ (𝑏𝑆𝑑 ∨ 𝑏 = 𝑑)))
32		neeq2 3027	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (⟨𝑎, 𝑏⟩ ≠ 𝑦 ↔ ⟨𝑎, 𝑏⟩ ≠ ⟨𝑐, 𝑑⟩))
33	6, 7	opthne 5465	. . . . 5 ⊢ (⟨𝑎, 𝑏⟩ ≠ ⟨𝑐, 𝑑⟩ ↔ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑑))
34	32, 33	bitrdi 290	. . . 4 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (⟨𝑎, 𝑏⟩ ≠ 𝑦 ↔ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑑)))
35	27, 31, 34	3anbi123d 1462	. . 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 1465	. 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 5529	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 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ⟨cop 4600   class class class wbr 5113  {copab 5177   × cxp 5660  ‘cfv 6537  1^st c1st 7983  2^nd c2nd 7984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fv 6545  df-1st 7985  df-2nd 7986

This theorem is referenced by:  poxp2  8138  frxp2  8139  xpord2pred  8140