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Theorem xporderlem 5793
Description: Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
Hypothesis
Ref Expression
xporderlem.1 𝑇 = {⟨x, y⟩ ∣ ((x (A × B) y (A × B)) ((1stx)𝑅(1sty) ((1stx) = (1sty) (2ndx)𝑆(2ndy))))}
Assertion
Ref Expression
xporderlem (⟨𝑎, 𝑏𝑇𝑐, 𝑑⟩ ↔ (((𝑎 A 𝑐 A) (𝑏 B 𝑑 B)) (𝑎𝑅𝑐 (𝑎 = 𝑐 𝑏𝑆𝑑))))
Distinct variable groups:   x,A,y   x,B,y   x,𝑅,y   x,𝑆,y   x,𝑎,y   x,𝑏,y   x,𝑐,y   x,𝑑,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 xporderlem
StepHypRef Expression
1 df-br 3756 . . 3 (⟨𝑎, 𝑏𝑇𝑐, 𝑑⟩ ↔ ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ 𝑇)
2 xporderlem.1 . . . 4 𝑇 = {⟨x, y⟩ ∣ ((x (A × B) y (A × B)) ((1stx)𝑅(1sty) ((1stx) = (1sty) (2ndx)𝑆(2ndy))))}
32eleq2i 2101 . . 3 (⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ 𝑇 ↔ ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ {⟨x, y⟩ ∣ ((x (A × B) y (A × B)) ((1stx)𝑅(1sty) ((1stx) = (1sty) (2ndx)𝑆(2ndy))))})
41, 3bitri 173 . 2 (⟨𝑎, 𝑏𝑇𝑐, 𝑑⟩ ↔ ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ {⟨x, y⟩ ∣ ((x (A × B) y (A × B)) ((1stx)𝑅(1sty) ((1stx) = (1sty) (2ndx)𝑆(2ndy))))})
5 vex 2554 . . . 4 𝑎 V
6 vex 2554 . . . 4 𝑏 V
75, 6opex 3957 . . 3 𝑎, 𝑏 V
8 vex 2554 . . . 4 𝑐 V
9 vex 2554 . . . 4 𝑑 V
108, 9opex 3957 . . 3 𝑐, 𝑑 V
11 eleq1 2097 . . . . . 6 (x = ⟨𝑎, 𝑏⟩ → (x (A × B) ↔ ⟨𝑎, 𝑏 (A × B)))
12 opelxp 4317 . . . . . 6 (⟨𝑎, 𝑏 (A × B) ↔ (𝑎 A 𝑏 B))
1311, 12syl6bb 185 . . . . 5 (x = ⟨𝑎, 𝑏⟩ → (x (A × B) ↔ (𝑎 A 𝑏 B)))
1413anbi1d 438 . . . 4 (x = ⟨𝑎, 𝑏⟩ → ((x (A × B) y (A × B)) ↔ ((𝑎 A 𝑏 B) y (A × B))))
155, 6op1std 5717 . . . . . 6 (x = ⟨𝑎, 𝑏⟩ → (1stx) = 𝑎)
1615breq1d 3765 . . . . 5 (x = ⟨𝑎, 𝑏⟩ → ((1stx)𝑅(1sty) ↔ 𝑎𝑅(1sty)))
1715eqeq1d 2045 . . . . . 6 (x = ⟨𝑎, 𝑏⟩ → ((1stx) = (1sty) ↔ 𝑎 = (1sty)))
185, 6op2ndd 5718 . . . . . . 7 (x = ⟨𝑎, 𝑏⟩ → (2ndx) = 𝑏)
1918breq1d 3765 . . . . . 6 (x = ⟨𝑎, 𝑏⟩ → ((2ndx)𝑆(2ndy) ↔ 𝑏𝑆(2ndy)))
2017, 19anbi12d 442 . . . . 5 (x = ⟨𝑎, 𝑏⟩ → (((1stx) = (1sty) (2ndx)𝑆(2ndy)) ↔ (𝑎 = (1sty) 𝑏𝑆(2ndy))))
2116, 20orbi12d 706 . . . 4 (x = ⟨𝑎, 𝑏⟩ → (((1stx)𝑅(1sty) ((1stx) = (1sty) (2ndx)𝑆(2ndy))) ↔ (𝑎𝑅(1sty) (𝑎 = (1sty) 𝑏𝑆(2ndy)))))
2214, 21anbi12d 442 . . 3 (x = ⟨𝑎, 𝑏⟩ → (((x (A × B) y (A × B)) ((1stx)𝑅(1sty) ((1stx) = (1sty) (2ndx)𝑆(2ndy)))) ↔ (((𝑎 A 𝑏 B) y (A × B)) (𝑎𝑅(1sty) (𝑎 = (1sty) 𝑏𝑆(2ndy))))))
23 eleq1 2097 . . . . . 6 (y = ⟨𝑐, 𝑑⟩ → (y (A × B) ↔ ⟨𝑐, 𝑑 (A × B)))
24 opelxp 4317 . . . . . 6 (⟨𝑐, 𝑑 (A × B) ↔ (𝑐 A 𝑑 B))
2523, 24syl6bb 185 . . . . 5 (y = ⟨𝑐, 𝑑⟩ → (y (A × B) ↔ (𝑐 A 𝑑 B)))
2625anbi2d 437 . . . 4 (y = ⟨𝑐, 𝑑⟩ → (((𝑎 A 𝑏 B) y (A × B)) ↔ ((𝑎 A 𝑏 B) (𝑐 A 𝑑 B))))
278, 9op1std 5717 . . . . . 6 (y = ⟨𝑐, 𝑑⟩ → (1sty) = 𝑐)
2827breq2d 3767 . . . . 5 (y = ⟨𝑐, 𝑑⟩ → (𝑎𝑅(1sty) ↔ 𝑎𝑅𝑐))
2927eqeq2d 2048 . . . . . 6 (y = ⟨𝑐, 𝑑⟩ → (𝑎 = (1sty) ↔ 𝑎 = 𝑐))
308, 9op2ndd 5718 . . . . . . 7 (y = ⟨𝑐, 𝑑⟩ → (2ndy) = 𝑑)
3130breq2d 3767 . . . . . 6 (y = ⟨𝑐, 𝑑⟩ → (𝑏𝑆(2ndy) ↔ 𝑏𝑆𝑑))
3229, 31anbi12d 442 . . . . 5 (y = ⟨𝑐, 𝑑⟩ → ((𝑎 = (1sty) 𝑏𝑆(2ndy)) ↔ (𝑎 = 𝑐 𝑏𝑆𝑑)))
3328, 32orbi12d 706 . . . 4 (y = ⟨𝑐, 𝑑⟩ → ((𝑎𝑅(1sty) (𝑎 = (1sty) 𝑏𝑆(2ndy))) ↔ (𝑎𝑅𝑐 (𝑎 = 𝑐 𝑏𝑆𝑑))))
3426, 33anbi12d 442 . . 3 (y = ⟨𝑐, 𝑑⟩ → ((((𝑎 A 𝑏 B) y (A × B)) (𝑎𝑅(1sty) (𝑎 = (1sty) 𝑏𝑆(2ndy)))) ↔ (((𝑎 A 𝑏 B) (𝑐 A 𝑑 B)) (𝑎𝑅𝑐 (𝑎 = 𝑐 𝑏𝑆𝑑)))))
357, 10, 22, 34opelopab 3999 . 2 (⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ {⟨x, y⟩ ∣ ((x (A × B) y (A × B)) ((1stx)𝑅(1sty) ((1stx) = (1sty) (2ndx)𝑆(2ndy))))} ↔ (((𝑎 A 𝑏 B) (𝑐 A 𝑑 B)) (𝑎𝑅𝑐 (𝑎 = 𝑐 𝑏𝑆𝑑))))
36 an4 520 . . 3 (((𝑎 A 𝑏 B) (𝑐 A 𝑑 B)) ↔ ((𝑎 A 𝑐 A) (𝑏 B 𝑑 B)))
3736anbi1i 431 . 2 ((((𝑎 A 𝑏 B) (𝑐 A 𝑑 B)) (𝑎𝑅𝑐 (𝑎 = 𝑐 𝑏𝑆𝑑))) ↔ (((𝑎 A 𝑐 A) (𝑏 B 𝑑 B)) (𝑎𝑅𝑐 (𝑎 = 𝑐 𝑏𝑆𝑑))))
384, 35, 373bitri 195 1 (⟨𝑎, 𝑏𝑇𝑐, 𝑑⟩ ↔ (((𝑎 A 𝑐 A) (𝑏 B 𝑑 B)) (𝑎𝑅𝑐 (𝑎 = 𝑐 𝑏𝑆𝑑))))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wo 628   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755  {copab 3808   × cxp 4286  cfv 4845  1st c1st 5707  2nd c2nd 5708
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-bnd 1396  ax-4 1397  ax-13 1401  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  ax-un 4136
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-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  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-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by:  poxp  5794
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