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Theorem oprabid 5457
Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable constraint between x, y, and z, we use ax-bnd 1376 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)
Assertion
Ref Expression
oprabid (⟨⟨x, y⟩, z {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ φ)

Proof of Theorem oprabid
Dummy variables 𝑎 𝑟 𝑠 𝑡 w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2534 . . . 4 x V
2 vex 2534 . . . 4 y V
31, 2opex 3936 . . 3 x, y V
4 vex 2534 . . 3 z V
5 opexg 3934 . . 3 ((⟨x, y V z V) → ⟨⟨x, y⟩, z V)
63, 4, 5mp2an 404 . 2 ⟨⟨x, y⟩, z V
73, 4eqvinop 3950 . . . . 5 (w = ⟨⟨x, y⟩, z⟩ ↔ 𝑎𝑡(w = ⟨𝑎, 𝑡𝑎, 𝑡⟩ = ⟨⟨x, y⟩, z⟩))
87biimpi 113 . . . 4 (w = ⟨⟨x, y⟩, z⟩ → 𝑎𝑡(w = ⟨𝑎, 𝑡𝑎, 𝑡⟩ = ⟨⟨x, y⟩, z⟩))
9 eqeq1 2024 . . . . . . . 8 (w = ⟨𝑎, 𝑡⟩ → (w = ⟨⟨x, y⟩, z⟩ ↔ ⟨𝑎, 𝑡⟩ = ⟨⟨x, y⟩, z⟩))
10 vex 2534 . . . . . . . . 9 𝑎 V
11 vex 2534 . . . . . . . . 9 𝑡 V
1210, 11opth1 3943 . . . . . . . 8 (⟨𝑎, 𝑡⟩ = ⟨⟨x, y⟩, z⟩ → 𝑎 = ⟨x, y⟩)
139, 12syl6bi 152 . . . . . . 7 (w = ⟨𝑎, 𝑡⟩ → (w = ⟨⟨x, y⟩, z⟩ → 𝑎 = ⟨x, y⟩))
141, 2eqvinop 3950 . . . . . . . . 9 (𝑎 = ⟨x, y⟩ ↔ 𝑟𝑠(𝑎 = ⟨𝑟, 𝑠𝑟, 𝑠⟩ = ⟨x, y⟩))
15 opeq1 3519 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑟, 𝑠⟩ → ⟨𝑎, 𝑡⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩)
1615eqeq2d 2029 . . . . . . . . . . . 12 (𝑎 = ⟨𝑟, 𝑠⟩ → (w = ⟨𝑎, 𝑡⟩ ↔ w = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩))
171, 2, 4otth2 3948 . . . . . . . . . . . . . . . . . . 19 (⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ↔ (x = 𝑟 y = 𝑠 z = 𝑡))
18 df-3an 873 . . . . . . . . . . . . . . . . . . 19 ((x = 𝑟 y = 𝑠 z = 𝑡) ↔ ((x = 𝑟 y = 𝑠) z = 𝑡))
1917, 18bitri 173 . . . . . . . . . . . . . . . . . 18 (⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ↔ ((x = 𝑟 y = 𝑠) z = 𝑡))
2019anbi1i 434 . . . . . . . . . . . . . . . . 17 ((⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡 φ) ↔ (((x = 𝑟 y = 𝑠) z = 𝑡) φ))
21 anass 383 . . . . . . . . . . . . . . . . 17 ((((x = 𝑟 y = 𝑠) z = 𝑡) φ) ↔ ((x = 𝑟 y = 𝑠) (z = 𝑡 φ)))
22 anass 383 . . . . . . . . . . . . . . . . 17 (((x = 𝑟 y = 𝑠) (z = 𝑡 φ)) ↔ (x = 𝑟 (y = 𝑠 (z = 𝑡 φ))))
2320, 21, 223bitri 195 . . . . . . . . . . . . . . . 16 ((⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡 φ) ↔ (x = 𝑟 (y = 𝑠 (z = 𝑡 φ))))
24233exbii 1476 . . . . . . . . . . . . . . 15 (xyz(⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡 φ) ↔ xyz(x = 𝑟 (y = 𝑠 (z = 𝑡 φ))))
25 oprabidlem 5456 . . . . . . . . . . . . . . . . . 18 (xz(x = 𝑟 (y = 𝑠 (z = 𝑡 φ))) → x(x = 𝑟 z(y = 𝑠 (z = 𝑡 φ))))
2625eximi 1469 . . . . . . . . . . . . . . . . 17 (yxz(x = 𝑟 (y = 𝑠 (z = 𝑡 φ))) → yx(x = 𝑟 z(y = 𝑠 (z = 𝑡 φ))))
27 excom 1532 . . . . . . . . . . . . . . . . 17 (xyz(x = 𝑟 (y = 𝑠 (z = 𝑡 φ))) ↔ yxz(x = 𝑟 (y = 𝑠 (z = 𝑡 φ))))
28 excom 1532 . . . . . . . . . . . . . . . . 17 (xy(x = 𝑟 z(y = 𝑠 (z = 𝑡 φ))) ↔ yx(x = 𝑟 z(y = 𝑠 (z = 𝑡 φ))))
2926, 27, 283imtr4i 190 . . . . . . . . . . . . . . . 16 (xyz(x = 𝑟 (y = 𝑠 (z = 𝑡 φ))) → xy(x = 𝑟 z(y = 𝑠 (z = 𝑡 φ))))
30 oprabidlem 5456 . . . . . . . . . . . . . . . 16 (xy(x = 𝑟 z(y = 𝑠 (z = 𝑡 φ))) → x(x = 𝑟 yz(y = 𝑠 (z = 𝑡 φ))))
31 oprabidlem 5456 . . . . . . . . . . . . . . . . . 18 (yz(y = 𝑠 (z = 𝑡 φ)) → y(y = 𝑠 z(z = 𝑡 φ)))
3231anim2i 324 . . . . . . . . . . . . . . . . 17 ((x = 𝑟 yz(y = 𝑠 (z = 𝑡 φ))) → (x = 𝑟 y(y = 𝑠 z(z = 𝑡 φ))))
3332eximi 1469 . . . . . . . . . . . . . . . 16 (x(x = 𝑟 yz(y = 𝑠 (z = 𝑡 φ))) → x(x = 𝑟 y(y = 𝑠 z(z = 𝑡 φ))))
3429, 30, 333syl 17 . . . . . . . . . . . . . . 15 (xyz(x = 𝑟 (y = 𝑠 (z = 𝑡 φ))) → x(x = 𝑟 y(y = 𝑠 z(z = 𝑡 φ))))
3524, 34sylbi 114 . . . . . . . . . . . . . 14 (xyz(⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡 φ) → x(x = 𝑟 y(y = 𝑠 z(z = 𝑡 φ))))
36 euequ1 1973 . . . . . . . . . . . . . . . . . . 19 ∃!x x = 𝑟
37 eupick 1957 . . . . . . . . . . . . . . . . . . 19 ((∃!x x = 𝑟 x(x = 𝑟 y(y = 𝑠 z(z = 𝑡 φ)))) → (x = 𝑟y(y = 𝑠 z(z = 𝑡 φ))))
3836, 37mpan 402 . . . . . . . . . . . . . . . . . 18 (x(x = 𝑟 y(y = 𝑠 z(z = 𝑡 φ))) → (x = 𝑟y(y = 𝑠 z(z = 𝑡 φ))))
39 euequ1 1973 . . . . . . . . . . . . . . . . . . . 20 ∃!y y = 𝑠
40 eupick 1957 . . . . . . . . . . . . . . . . . . . 20 ((∃!y y = 𝑠 y(y = 𝑠 z(z = 𝑡 φ))) → (y = 𝑠z(z = 𝑡 φ)))
4139, 40mpan 402 . . . . . . . . . . . . . . . . . . 19 (y(y = 𝑠 z(z = 𝑡 φ)) → (y = 𝑠z(z = 𝑡 φ)))
42 euequ1 1973 . . . . . . . . . . . . . . . . . . . 20 ∃!z z = 𝑡
43 eupick 1957 . . . . . . . . . . . . . . . . . . . 20 ((∃!z z = 𝑡 z(z = 𝑡 φ)) → (z = 𝑡φ))
4442, 43mpan 402 . . . . . . . . . . . . . . . . . . 19 (z(z = 𝑡 φ) → (z = 𝑡φ))
4541, 44syl6 29 . . . . . . . . . . . . . . . . . 18 (y(y = 𝑠 z(z = 𝑡 φ)) → (y = 𝑠 → (z = 𝑡φ)))
4638, 45syl6 29 . . . . . . . . . . . . . . . . 17 (x(x = 𝑟 y(y = 𝑠 z(z = 𝑡 φ))) → (x = 𝑟 → (y = 𝑠 → (z = 𝑡φ))))
47463impd 1102 . . . . . . . . . . . . . . . 16 (x(x = 𝑟 y(y = 𝑠 z(z = 𝑡 φ))) → ((x = 𝑟 y = 𝑠 z = 𝑡) → φ))
4817, 47syl5bi 141 . . . . . . . . . . . . . . 15 (x(x = 𝑟 y(y = 𝑠 z(z = 𝑡 φ))) → (⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → φ))
4948com12 27 . . . . . . . . . . . . . 14 (⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (x(x = 𝑟 y(y = 𝑠 z(z = 𝑡 φ))) → φ))
5035, 49syl5 28 . . . . . . . . . . . . 13 (⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (xyz(⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡 φ) → φ))
51 eqeq1 2024 . . . . . . . . . . . . . . 15 (w = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (w = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ = ⟨⟨x, y⟩, z⟩))
52 eqcom 2020 . . . . . . . . . . . . . . 15 (⟨⟨𝑟, 𝑠⟩, 𝑡⟩ = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩)
5351, 52syl6bb 185 . . . . . . . . . . . . . 14 (w = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (w = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩))
5453anbi1d 441 . . . . . . . . . . . . . . . 16 (w = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → ((w = ⟨⟨x, y⟩, z φ) ↔ (⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡 φ)))
55543exbidv 1727 . . . . . . . . . . . . . . 15 (w = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) ↔ xyz(⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡 φ)))
5655imbi1d 220 . . . . . . . . . . . . . 14 (w = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → ((xyz(w = ⟨⟨x, y⟩, z φ) → φ) ↔ (xyz(⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡 φ) → φ)))
5753, 56imbi12d 223 . . . . . . . . . . . . 13 (w = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → ((w = ⟨⟨x, y⟩, z⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) → φ)) ↔ (⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (xyz(⟨⟨x, y⟩, z⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡 φ) → φ))))
5850, 57mpbiri 157 . . . . . . . . . . . 12 (w = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (w = ⟨⟨x, y⟩, z⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) → φ)))
5916, 58syl6bi 152 . . . . . . . . . . 11 (𝑎 = ⟨𝑟, 𝑠⟩ → (w = ⟨𝑎, 𝑡⟩ → (w = ⟨⟨x, y⟩, z⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) → φ))))
6059adantr 261 . . . . . . . . . 10 ((𝑎 = ⟨𝑟, 𝑠𝑟, 𝑠⟩ = ⟨x, y⟩) → (w = ⟨𝑎, 𝑡⟩ → (w = ⟨⟨x, y⟩, z⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) → φ))))
6160exlimivv 1754 . . . . . . . . 9 (𝑟𝑠(𝑎 = ⟨𝑟, 𝑠𝑟, 𝑠⟩ = ⟨x, y⟩) → (w = ⟨𝑎, 𝑡⟩ → (w = ⟨⟨x, y⟩, z⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) → φ))))
6214, 61sylbi 114 . . . . . . . 8 (𝑎 = ⟨x, y⟩ → (w = ⟨𝑎, 𝑡⟩ → (w = ⟨⟨x, y⟩, z⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) → φ))))
6362com3l 75 . . . . . . 7 (w = ⟨𝑎, 𝑡⟩ → (w = ⟨⟨x, y⟩, z⟩ → (𝑎 = ⟨x, y⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) → φ))))
6413, 63mpdd 36 . . . . . 6 (w = ⟨𝑎, 𝑡⟩ → (w = ⟨⟨x, y⟩, z⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) → φ)))
6564adantr 261 . . . . 5 ((w = ⟨𝑎, 𝑡𝑎, 𝑡⟩ = ⟨⟨x, y⟩, z⟩) → (w = ⟨⟨x, y⟩, z⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) → φ)))
6665exlimivv 1754 . . . 4 (𝑎𝑡(w = ⟨𝑎, 𝑡𝑎, 𝑡⟩ = ⟨⟨x, y⟩, z⟩) → (w = ⟨⟨x, y⟩, z⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) → φ)))
678, 66mpcom 32 . . 3 (w = ⟨⟨x, y⟩, z⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) → φ))
68 19.8a 1460 . . . . 5 ((w = ⟨⟨x, y⟩, z φ) → z(w = ⟨⟨x, y⟩, z φ))
69 19.8a 1460 . . . . 5 (z(w = ⟨⟨x, y⟩, z φ) → yz(w = ⟨⟨x, y⟩, z φ))
70 19.8a 1460 . . . . 5 (yz(w = ⟨⟨x, y⟩, z φ) → xyz(w = ⟨⟨x, y⟩, z φ))
7168, 69, 703syl 17 . . . 4 ((w = ⟨⟨x, y⟩, z φ) → xyz(w = ⟨⟨x, y⟩, z φ))
7271ex 108 . . 3 (w = ⟨⟨x, y⟩, z⟩ → (φxyz(w = ⟨⟨x, y⟩, z φ)))
7367, 72impbid 120 . 2 (w = ⟨⟨x, y⟩, z⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) ↔ φ))
74 df-oprab 5436 . 2 {⟨⟨x, y⟩, z⟩ ∣ φ} = {wxyz(w = ⟨⟨x, y⟩, z φ)}
756, 73, 74elab2 2663 1 (⟨⟨x, y⟩, z {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 871   = wceq 1226  wex 1358   wcel 1370  ∃!weu 1878  Vcvv 2531  cop 3349  {coprab 5433
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-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-setind 4200
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-v 2533  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-oprab 5436
This theorem is referenced by:  ssoprab2b  5481  ovid  5536  ovidig  5537  tposoprab  5813
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