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Theorem brrestrict 33296
Description: Binary relation form of the Restrict function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brrestrict.1 𝐴 ∈ V
brrestrict.2 𝐵 ∈ V
brrestrict.3 𝐶 ∈ V
Assertion
Ref Expression
brrestrict (⟨𝐴, 𝐵⟩Restrict𝐶𝐶 = (𝐴𝐵))

Proof of Theorem brrestrict
Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 5352 . . . . 5 𝐴, 𝐵⟩ ∈ V
2 brrestrict.3 . . . . 5 𝐶 ∈ V
31, 2brco 5739 . . . 4 (⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶 ↔ ∃𝑥(⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥Cap𝐶))
41brtxp2 33228 . . . . . . 7 (⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥 ↔ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩1st 𝑎 ∧ ⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏))
5 3anrot 1094 . . . . . . . . 9 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩1st 𝑎 ∧ ⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏) ↔ (⟨𝐴, 𝐵⟩1st 𝑎 ∧ ⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏𝑥 = ⟨𝑎, 𝑏⟩))
6 brrestrict.1 . . . . . . . . . . 11 𝐴 ∈ V
7 brrestrict.2 . . . . . . . . . . 11 𝐵 ∈ V
86, 7br1steq 32900 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩1st 𝑎𝑎 = 𝐴)
9 vex 3502 . . . . . . . . . . . 12 𝑏 ∈ V
101, 9brco 5739 . . . . . . . . . . 11 (⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏 ↔ ∃𝑥(⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏))
111brtxp2 33228 . . . . . . . . . . . . . . 15 (⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥 ↔ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏))
12 3anrot 1094 . . . . . . . . . . . . . . . . 17 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏) ↔ (⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏𝑥 = ⟨𝑎, 𝑏⟩))
136, 7br2ndeq 32901 . . . . . . . . . . . . . . . . . 18 (⟨𝐴, 𝐵⟩2nd 𝑎𝑎 = 𝐵)
141, 9brco 5739 . . . . . . . . . . . . . . . . . . 19 (⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏 ↔ ∃𝑥(⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏))
156, 7br1steq 32900 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝐴, 𝐵⟩1st 𝑥𝑥 = 𝐴)
1615anbi1i 623 . . . . . . . . . . . . . . . . . . . . 21 ((⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏) ↔ (𝑥 = 𝐴𝑥Range𝑏))
1716exbii 1841 . . . . . . . . . . . . . . . . . . . 20 (∃𝑥(⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏) ↔ ∃𝑥(𝑥 = 𝐴𝑥Range𝑏))
18 breq1 5065 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝐴 → (𝑥Range𝑏𝐴Range𝑏))
196, 18ceqsexv 3546 . . . . . . . . . . . . . . . . . . . 20 (∃𝑥(𝑥 = 𝐴𝑥Range𝑏) ↔ 𝐴Range𝑏)
2017, 19bitri 276 . . . . . . . . . . . . . . . . . . 19 (∃𝑥(⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏) ↔ 𝐴Range𝑏)
216, 9brrange 33281 . . . . . . . . . . . . . . . . . . 19 (𝐴Range𝑏𝑏 = ran 𝐴)
2214, 20, 213bitri 298 . . . . . . . . . . . . . . . . . 18 (⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏𝑏 = ran 𝐴)
23 biid 262 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝑎, 𝑏⟩)
2413, 22, 233anbi123i 1149 . . . . . . . . . . . . . . . . 17 ((⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏𝑥 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩))
2512, 24bitri 276 . . . . . . . . . . . . . . . 16 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏) ↔ (𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩))
26252exbii 1842 . . . . . . . . . . . . . . 15 (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏) ↔ ∃𝑎𝑏(𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩))
276rnex 7608 . . . . . . . . . . . . . . . 16 ran 𝐴 ∈ V
28 opeq1 4801 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝐵 → ⟨𝑎, 𝑏⟩ = ⟨𝐵, 𝑏⟩)
2928eqeq2d 2836 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐵 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐵, 𝑏⟩))
30 opeq2 4802 . . . . . . . . . . . . . . . . 17 (𝑏 = ran 𝐴 → ⟨𝐵, 𝑏⟩ = ⟨𝐵, ran 𝐴⟩)
3130eqeq2d 2836 . . . . . . . . . . . . . . . 16 (𝑏 = ran 𝐴 → (𝑥 = ⟨𝐵, 𝑏⟩ ↔ 𝑥 = ⟨𝐵, ran 𝐴⟩))
327, 27, 29, 31ceqsex2v 3549 . . . . . . . . . . . . . . 15 (∃𝑎𝑏(𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩) ↔ 𝑥 = ⟨𝐵, ran 𝐴⟩)
3311, 26, 323bitri 298 . . . . . . . . . . . . . 14 (⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥 = ⟨𝐵, ran 𝐴⟩)
3433anbi1i 623 . . . . . . . . . . . . 13 ((⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏) ↔ (𝑥 = ⟨𝐵, ran 𝐴⟩ ∧ 𝑥Cart𝑏))
3534exbii 1841 . . . . . . . . . . . 12 (∃𝑥(⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏) ↔ ∃𝑥(𝑥 = ⟨𝐵, ran 𝐴⟩ ∧ 𝑥Cart𝑏))
36 opex 5352 . . . . . . . . . . . . 13 𝐵, ran 𝐴⟩ ∈ V
37 breq1 5065 . . . . . . . . . . . . 13 (𝑥 = ⟨𝐵, ran 𝐴⟩ → (𝑥Cart𝑏 ↔ ⟨𝐵, ran 𝐴⟩Cart𝑏))
3836, 37ceqsexv 3546 . . . . . . . . . . . 12 (∃𝑥(𝑥 = ⟨𝐵, ran 𝐴⟩ ∧ 𝑥Cart𝑏) ↔ ⟨𝐵, ran 𝐴⟩Cart𝑏)
3935, 38bitri 276 . . . . . . . . . . 11 (∃𝑥(⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏) ↔ ⟨𝐵, ran 𝐴⟩Cart𝑏)
407, 27, 9brcart 33279 . . . . . . . . . . 11 (⟨𝐵, ran 𝐴⟩Cart𝑏𝑏 = (𝐵 × ran 𝐴))
4110, 39, 403bitri 298 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏𝑏 = (𝐵 × ran 𝐴))
428, 41, 233anbi123i 1149 . . . . . . . . 9 ((⟨𝐴, 𝐵⟩1st 𝑎 ∧ ⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏𝑥 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝐴𝑏 = (𝐵 × ran 𝐴) ∧ 𝑥 = ⟨𝑎, 𝑏⟩))
435, 42bitri 276 . . . . . . . 8 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩1st 𝑎 ∧ ⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏) ↔ (𝑎 = 𝐴𝑏 = (𝐵 × ran 𝐴) ∧ 𝑥 = ⟨𝑎, 𝑏⟩))
44432exbii 1842 . . . . . . 7 (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩1st 𝑎 ∧ ⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏) ↔ ∃𝑎𝑏(𝑎 = 𝐴𝑏 = (𝐵 × ran 𝐴) ∧ 𝑥 = ⟨𝑎, 𝑏⟩))
457, 27xpex 7468 . . . . . . . 8 (𝐵 × ran 𝐴) ∈ V
46 opeq1 4801 . . . . . . . . 9 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
4746eqeq2d 2836 . . . . . . . 8 (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
48 opeq2 4802 . . . . . . . . 9 (𝑏 = (𝐵 × ran 𝐴) → ⟨𝐴, 𝑏⟩ = ⟨𝐴, (𝐵 × ran 𝐴)⟩)
4948eqeq2d 2836 . . . . . . . 8 (𝑏 = (𝐵 × ran 𝐴) → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩))
506, 45, 47, 49ceqsex2v 3549 . . . . . . 7 (∃𝑎𝑏(𝑎 = 𝐴𝑏 = (𝐵 × ran 𝐴) ∧ 𝑥 = ⟨𝑎, 𝑏⟩) ↔ 𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩)
514, 44, 503bitri 298 . . . . . 6 (⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩)
5251anbi1i 623 . . . . 5 ((⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥Cap𝐶) ↔ (𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶))
5352exbii 1841 . . . 4 (∃𝑥(⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥Cap𝐶) ↔ ∃𝑥(𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶))
543, 53bitri 276 . . 3 (⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶 ↔ ∃𝑥(𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶))
55 opex 5352 . . . 4 𝐴, (𝐵 × ran 𝐴)⟩ ∈ V
56 breq1 5065 . . . 4 (𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ → (𝑥Cap𝐶 ↔ ⟨𝐴, (𝐵 × ran 𝐴)⟩Cap𝐶))
5755, 56ceqsexv 3546 . . 3 (∃𝑥(𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶) ↔ ⟨𝐴, (𝐵 × ran 𝐴)⟩Cap𝐶)
586, 45, 2brcap 33287 . . 3 (⟨𝐴, (𝐵 × ran 𝐴)⟩Cap𝐶𝐶 = (𝐴 ∩ (𝐵 × ran 𝐴)))
5954, 57, 583bitri 298 . 2 (⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶𝐶 = (𝐴 ∩ (𝐵 × ran 𝐴)))
60 df-restrict 33218 . . 3 Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
6160breqi 5068 . 2 (⟨𝐴, 𝐵⟩Restrict𝐶 ↔ ⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶)
62 dfres3 5856 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))
6362eqeq2i 2838 . 2 (𝐶 = (𝐴𝐵) ↔ 𝐶 = (𝐴 ∩ (𝐵 × ran 𝐴)))
6459, 61, 633bitr4i 304 1 (⟨𝐴, 𝐵⟩Restrict𝐶𝐶 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  Vcvv 3499  cin 3938  cop 4569   class class class wbr 5062   × cxp 5551  ran crn 5554  cres 5555  ccom 5557  1st c1st 7681  2nd c2nd 7682  ctxp 33177  Cartccart 33188  Rangecrange 33191  Capccap 33194  Restrictcrestrict 33198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-symdif 4222  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-eprel 5463  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fo 6357  df-fv 6359  df-1st 7683  df-2nd 7684  df-txp 33201  df-pprod 33202  df-image 33211  df-cart 33212  df-range 33215  df-cap 33217  df-restrict 33218
This theorem is referenced by:  dfrecs2  33297
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