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Theorem brrestrict 31751
Description: The binary relationship 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 4903 . . . . 5 𝐴, 𝐵⟩ ∈ V
2 brrestrict.3 . . . . 5 𝐶 ∈ V
31, 2brco 5262 . . . 4 (⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶 ↔ ∃𝑥(⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥Cap𝐶))
41brtxp2 31683 . . . . . . 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 1041 . . . . . . . . 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
8 vex 3193 . . . . . . . . . . 11 𝑎 ∈ V
96, 7, 8br1steq 31427 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩1st 𝑎𝑎 = 𝐴)
10 vex 3193 . . . . . . . . . . . 12 𝑏 ∈ V
111, 10brco 5262 . . . . . . . . . . 11 (⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏 ↔ ∃𝑥(⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏))
121brtxp2 31683 . . . . . . . . . . . . . . 15 (⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥 ↔ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏))
13 3anrot 1041 . . . . . . . . . . . . . . . . 17 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏) ↔ (⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏𝑥 = ⟨𝑎, 𝑏⟩))
146, 7, 8br2ndeq 31428 . . . . . . . . . . . . . . . . . 18 (⟨𝐴, 𝐵⟩2nd 𝑎𝑎 = 𝐵)
151, 10brco 5262 . . . . . . . . . . . . . . . . . . 19 (⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏 ↔ ∃𝑥(⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏))
16 vex 3193 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥 ∈ V
176, 7, 16br1steq 31427 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝐴, 𝐵⟩1st 𝑥𝑥 = 𝐴)
1817anbi1i 730 . . . . . . . . . . . . . . . . . . . . 21 ((⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏) ↔ (𝑥 = 𝐴𝑥Range𝑏))
1918exbii 1771 . . . . . . . . . . . . . . . . . . . 20 (∃𝑥(⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏) ↔ ∃𝑥(𝑥 = 𝐴𝑥Range𝑏))
20 breq1 4626 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝐴 → (𝑥Range𝑏𝐴Range𝑏))
216, 20ceqsexv 3232 . . . . . . . . . . . . . . . . . . . 20 (∃𝑥(𝑥 = 𝐴𝑥Range𝑏) ↔ 𝐴Range𝑏)
2219, 21bitri 264 . . . . . . . . . . . . . . . . . . 19 (∃𝑥(⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏) ↔ 𝐴Range𝑏)
236, 10brrange 31736 . . . . . . . . . . . . . . . . . . 19 (𝐴Range𝑏𝑏 = ran 𝐴)
2415, 22, 233bitri 286 . . . . . . . . . . . . . . . . . 18 (⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏𝑏 = ran 𝐴)
25 biid 251 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝑎, 𝑏⟩)
2614, 24, 253anbi123i 1249 . . . . . . . . . . . . . . . . 17 ((⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏𝑥 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩))
2713, 26bitri 264 . . . . . . . . . . . . . . . 16 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏) ↔ (𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩))
28272exbii 1772 . . . . . . . . . . . . . . 15 (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏) ↔ ∃𝑎𝑏(𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩))
296rnex 7062 . . . . . . . . . . . . . . . 16 ran 𝐴 ∈ V
30 opeq1 4377 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝐵 → ⟨𝑎, 𝑏⟩ = ⟨𝐵, 𝑏⟩)
3130eqeq2d 2631 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐵 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐵, 𝑏⟩))
32 opeq2 4378 . . . . . . . . . . . . . . . . 17 (𝑏 = ran 𝐴 → ⟨𝐵, 𝑏⟩ = ⟨𝐵, ran 𝐴⟩)
3332eqeq2d 2631 . . . . . . . . . . . . . . . 16 (𝑏 = ran 𝐴 → (𝑥 = ⟨𝐵, 𝑏⟩ ↔ 𝑥 = ⟨𝐵, ran 𝐴⟩))
347, 29, 31, 33ceqsex2v 3235 . . . . . . . . . . . . . . 15 (∃𝑎𝑏(𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩) ↔ 𝑥 = ⟨𝐵, ran 𝐴⟩)
3512, 28, 343bitri 286 . . . . . . . . . . . . . 14 (⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥 = ⟨𝐵, ran 𝐴⟩)
3635anbi1i 730 . . . . . . . . . . . . 13 ((⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏) ↔ (𝑥 = ⟨𝐵, ran 𝐴⟩ ∧ 𝑥Cart𝑏))
3736exbii 1771 . . . . . . . . . . . 12 (∃𝑥(⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏) ↔ ∃𝑥(𝑥 = ⟨𝐵, ran 𝐴⟩ ∧ 𝑥Cart𝑏))
38 opex 4903 . . . . . . . . . . . . 13 𝐵, ran 𝐴⟩ ∈ V
39 breq1 4626 . . . . . . . . . . . . 13 (𝑥 = ⟨𝐵, ran 𝐴⟩ → (𝑥Cart𝑏 ↔ ⟨𝐵, ran 𝐴⟩Cart𝑏))
4038, 39ceqsexv 3232 . . . . . . . . . . . 12 (∃𝑥(𝑥 = ⟨𝐵, ran 𝐴⟩ ∧ 𝑥Cart𝑏) ↔ ⟨𝐵, ran 𝐴⟩Cart𝑏)
4137, 40bitri 264 . . . . . . . . . . 11 (∃𝑥(⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏) ↔ ⟨𝐵, ran 𝐴⟩Cart𝑏)
427, 29, 10brcart 31734 . . . . . . . . . . 11 (⟨𝐵, ran 𝐴⟩Cart𝑏𝑏 = (𝐵 × ran 𝐴))
4311, 41, 423bitri 286 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏𝑏 = (𝐵 × ran 𝐴))
449, 43, 253anbi123i 1249 . . . . . . . . 9 ((⟨𝐴, 𝐵⟩1st 𝑎 ∧ ⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏𝑥 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝐴𝑏 = (𝐵 × ran 𝐴) ∧ 𝑥 = ⟨𝑎, 𝑏⟩))
455, 44bitri 264 . . . . . . . 8 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩1st 𝑎 ∧ ⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏) ↔ (𝑎 = 𝐴𝑏 = (𝐵 × ran 𝐴) ∧ 𝑥 = ⟨𝑎, 𝑏⟩))
46452exbii 1772 . . . . . . 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⟩))
477, 29xpex 6927 . . . . . . . 8 (𝐵 × ran 𝐴) ∈ V
48 opeq1 4377 . . . . . . . . 9 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
4948eqeq2d 2631 . . . . . . . 8 (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
50 opeq2 4378 . . . . . . . . 9 (𝑏 = (𝐵 × ran 𝐴) → ⟨𝐴, 𝑏⟩ = ⟨𝐴, (𝐵 × ran 𝐴)⟩)
5150eqeq2d 2631 . . . . . . . 8 (𝑏 = (𝐵 × ran 𝐴) → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩))
526, 47, 49, 51ceqsex2v 3235 . . . . . . 7 (∃𝑎𝑏(𝑎 = 𝐴𝑏 = (𝐵 × ran 𝐴) ∧ 𝑥 = ⟨𝑎, 𝑏⟩) ↔ 𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩)
534, 46, 523bitri 286 . . . . . 6 (⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩)
5453anbi1i 730 . . . . 5 ((⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥Cap𝐶) ↔ (𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶))
5554exbii 1771 . . . 4 (∃𝑥(⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥Cap𝐶) ↔ ∃𝑥(𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶))
563, 55bitri 264 . . 3 (⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶 ↔ ∃𝑥(𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶))
57 opex 4903 . . . 4 𝐴, (𝐵 × ran 𝐴)⟩ ∈ V
58 breq1 4626 . . . 4 (𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ → (𝑥Cap𝐶 ↔ ⟨𝐴, (𝐵 × ran 𝐴)⟩Cap𝐶))
5957, 58ceqsexv 3232 . . 3 (∃𝑥(𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶) ↔ ⟨𝐴, (𝐵 × ran 𝐴)⟩Cap𝐶)
606, 47, 2brcap 31742 . . 3 (⟨𝐴, (𝐵 × ran 𝐴)⟩Cap𝐶𝐶 = (𝐴 ∩ (𝐵 × ran 𝐴)))
6156, 59, 603bitri 286 . 2 (⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶𝐶 = (𝐴 ∩ (𝐵 × ran 𝐴)))
62 df-restrict 31672 . . 3 Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
6362breqi 4629 . 2 (⟨𝐴, 𝐵⟩Restrict𝐶 ↔ ⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶)
64 dfres3 31410 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))
6564eqeq2i 2633 . 2 (𝐶 = (𝐴𝐵) ↔ 𝐶 = (𝐴 ∩ (𝐵 × ran 𝐴)))
6661, 63, 653bitr4i 292 1 (⟨𝐴, 𝐵⟩Restrict𝐶𝐶 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  Vcvv 3190  cin 3559  cop 4161   class class class wbr 4623   × cxp 5082  ran crn 5085  cres 5086  ccom 5088  1st c1st 7126  2nd c2nd 7127  ctxp 31631  Cartccart 31642  Rangecrange 31645  Capccap 31648  Restrictcrestrict 31652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-symdif 3828  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-eprel 4995  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-fv 5865  df-1st 7128  df-2nd 7129  df-txp 31655  df-pprod 31656  df-image 31665  df-cart 31666  df-range 31669  df-cap 31671  df-restrict 31672
This theorem is referenced by:  dfrecs2  31752
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