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Theorem brrestrict 33308
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 5348 . . . . 5 𝐴, 𝐵⟩ ∈ V
2 brrestrict.3 . . . . 5 𝐶 ∈ V
31, 2brco 5735 . . . 4 (⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶 ↔ ∃𝑥(⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥Cap𝐶))
41brtxp2 33240 . . . . . . 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 1092 . . . . . . . . 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 32912 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩1st 𝑎𝑎 = 𝐴)
9 vex 3498 . . . . . . . . . . . 12 𝑏 ∈ V
101, 9brco 5735 . . . . . . . . . . 11 (⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏 ↔ ∃𝑥(⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏))
111brtxp2 33240 . . . . . . . . . . . . . . 15 (⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥 ↔ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏))
12 3anrot 1092 . . . . . . . . . . . . . . . . 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 32913 . . . . . . . . . . . . . . . . . 18 (⟨𝐴, 𝐵⟩2nd 𝑎𝑎 = 𝐵)
141, 9brco 5735 . . . . . . . . . . . . . . . . . . 19 (⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏 ↔ ∃𝑥(⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏))
156, 7br1steq 32912 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝐴, 𝐵⟩1st 𝑥𝑥 = 𝐴)
1615anbi1i 623 . . . . . . . . . . . . . . . . . . . . 21 ((⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏) ↔ (𝑥 = 𝐴𝑥Range𝑏))
1716exbii 1839 . . . . . . . . . . . . . . . . . . . 20 (∃𝑥(⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏) ↔ ∃𝑥(𝑥 = 𝐴𝑥Range𝑏))
18 breq1 5061 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝐴 → (𝑥Range𝑏𝐴Range𝑏))
196, 18ceqsexv 3542 . . . . . . . . . . . . . . . . . . . 20 (∃𝑥(𝑥 = 𝐴𝑥Range𝑏) ↔ 𝐴Range𝑏)
2017, 19bitri 276 . . . . . . . . . . . . . . . . . . 19 (∃𝑥(⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏) ↔ 𝐴Range𝑏)
216, 9brrange 33293 . . . . . . . . . . . . . . . . . . 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 1147 . . . . . . . . . . . . . . . . 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 1840 . . . . . . . . . . . . . . 15 (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏) ↔ ∃𝑎𝑏(𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩))
276rnex 7605 . . . . . . . . . . . . . . . 16 ran 𝐴 ∈ V
28 opeq1 4797 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝐵 → ⟨𝑎, 𝑏⟩ = ⟨𝐵, 𝑏⟩)
2928eqeq2d 2832 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐵 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐵, 𝑏⟩))
30 opeq2 4798 . . . . . . . . . . . . . . . . 17 (𝑏 = ran 𝐴 → ⟨𝐵, 𝑏⟩ = ⟨𝐵, ran 𝐴⟩)
3130eqeq2d 2832 . . . . . . . . . . . . . . . 16 (𝑏 = ran 𝐴 → (𝑥 = ⟨𝐵, 𝑏⟩ ↔ 𝑥 = ⟨𝐵, ran 𝐴⟩))
327, 27, 29, 31ceqsex2v 3545 . . . . . . . . . . . . . . 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 1839 . . . . . . . . . . . 12 (∃𝑥(⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏) ↔ ∃𝑥(𝑥 = ⟨𝐵, ran 𝐴⟩ ∧ 𝑥Cart𝑏))
36 opex 5348 . . . . . . . . . . . . 13 𝐵, ran 𝐴⟩ ∈ V
37 breq1 5061 . . . . . . . . . . . . 13 (𝑥 = ⟨𝐵, ran 𝐴⟩ → (𝑥Cart𝑏 ↔ ⟨𝐵, ran 𝐴⟩Cart𝑏))
3836, 37ceqsexv 3542 . . . . . . . . . . . 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 33291 . . . . . . . . . . 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 1147 . . . . . . . . 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 1840 . . . . . . 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 7464 . . . . . . . 8 (𝐵 × ran 𝐴) ∈ V
46 opeq1 4797 . . . . . . . . 9 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
4746eqeq2d 2832 . . . . . . . 8 (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
48 opeq2 4798 . . . . . . . . 9 (𝑏 = (𝐵 × ran 𝐴) → ⟨𝐴, 𝑏⟩ = ⟨𝐴, (𝐵 × ran 𝐴)⟩)
4948eqeq2d 2832 . . . . . . . 8 (𝑏 = (𝐵 × ran 𝐴) → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩))
506, 45, 47, 49ceqsex2v 3545 . . . . . . 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 1839 . . . 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 5348 . . . 4 𝐴, (𝐵 × ran 𝐴)⟩ ∈ V
56 breq1 5061 . . . 4 (𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ → (𝑥Cap𝐶 ↔ ⟨𝐴, (𝐵 × ran 𝐴)⟩Cap𝐶))
5755, 56ceqsexv 3542 . . 3 (∃𝑥(𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶) ↔ ⟨𝐴, (𝐵 × ran 𝐴)⟩Cap𝐶)
586, 45, 2brcap 33299 . . 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 33230 . . 3 Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
6160breqi 5064 . 2 (⟨𝐴, 𝐵⟩Restrict𝐶 ↔ ⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶)
62 dfres3 5852 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))
6362eqeq2i 2834 . 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 1079   = wceq 1528  wex 1771  wcel 2105  Vcvv 3495  cin 3934  cop 4565   class class class wbr 5058   × cxp 5547  ran crn 5550  cres 5551  ccom 5553  1st c1st 7678  2nd c2nd 7679  ctxp 33189  Cartccart 33200  Rangecrange 33203  Capccap 33206  Restrictcrestrict 33210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-symdif 4218  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-eprel 5459  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fo 6355  df-fv 6357  df-1st 7680  df-2nd 7681  df-txp 33213  df-pprod 33214  df-image 33223  df-cart 33224  df-range 33227  df-cap 33229  df-restrict 33230
This theorem is referenced by:  dfrecs2  33309
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