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Theorem brrestrict 32362
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 5081 . . . . 5 𝐴, 𝐵⟩ ∈ V
2 brrestrict.3 . . . . 5 𝐶 ∈ V
31, 2brco 5448 . . . 4 (⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶 ↔ ∃𝑥(⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥Cap𝐶))
41brtxp2 32294 . . . . . . 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 1087 . . . . . . . . 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 31977 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩1st 𝑎𝑎 = 𝐴)
9 vex 3343 . . . . . . . . . . . 12 𝑏 ∈ V
101, 9brco 5448 . . . . . . . . . . 11 (⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏 ↔ ∃𝑥(⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏))
111brtxp2 32294 . . . . . . . . . . . . . . 15 (⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥 ↔ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏))
12 3anrot 1087 . . . . . . . . . . . . . . . . 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 31978 . . . . . . . . . . . . . . . . . 18 (⟨𝐴, 𝐵⟩2nd 𝑎𝑎 = 𝐵)
141, 9brco 5448 . . . . . . . . . . . . . . . . . . 19 (⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏 ↔ ∃𝑥(⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏))
156, 7br1steq 31977 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝐴, 𝐵⟩1st 𝑥𝑥 = 𝐴)
1615anbi1i 733 . . . . . . . . . . . . . . . . . . . . 21 ((⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏) ↔ (𝑥 = 𝐴𝑥Range𝑏))
1716exbii 1923 . . . . . . . . . . . . . . . . . . . 20 (∃𝑥(⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏) ↔ ∃𝑥(𝑥 = 𝐴𝑥Range𝑏))
18 breq1 4807 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝐴 → (𝑥Range𝑏𝐴Range𝑏))
196, 18ceqsexv 3382 . . . . . . . . . . . . . . . . . . . 20 (∃𝑥(𝑥 = 𝐴𝑥Range𝑏) ↔ 𝐴Range𝑏)
2017, 19bitri 264 . . . . . . . . . . . . . . . . . . 19 (∃𝑥(⟨𝐴, 𝐵⟩1st 𝑥𝑥Range𝑏) ↔ 𝐴Range𝑏)
216, 9brrange 32347 . . . . . . . . . . . . . . . . . . 19 (𝐴Range𝑏𝑏 = ran 𝐴)
2214, 20, 213bitri 286 . . . . . . . . . . . . . . . . . 18 (⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏𝑏 = ran 𝐴)
23 biid 251 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝑎, 𝑏⟩)
2413, 22, 233anbi123i 1159 . . . . . . . . . . . . . . . . 17 ((⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏𝑥 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩))
2512, 24bitri 264 . . . . . . . . . . . . . . . 16 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏) ↔ (𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩))
26252exbii 1924 . . . . . . . . . . . . . . 15 (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩2nd 𝑎 ∧ ⟨𝐴, 𝐵⟩(Range ∘ 1st )𝑏) ↔ ∃𝑎𝑏(𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩))
276rnex 7265 . . . . . . . . . . . . . . . 16 ran 𝐴 ∈ V
28 opeq1 4553 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝐵 → ⟨𝑎, 𝑏⟩ = ⟨𝐵, 𝑏⟩)
2928eqeq2d 2770 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐵 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐵, 𝑏⟩))
30 opeq2 4554 . . . . . . . . . . . . . . . . 17 (𝑏 = ran 𝐴 → ⟨𝐵, 𝑏⟩ = ⟨𝐵, ran 𝐴⟩)
3130eqeq2d 2770 . . . . . . . . . . . . . . . 16 (𝑏 = ran 𝐴 → (𝑥 = ⟨𝐵, 𝑏⟩ ↔ 𝑥 = ⟨𝐵, ran 𝐴⟩))
327, 27, 29, 31ceqsex2v 3385 . . . . . . . . . . . . . . 15 (∃𝑎𝑏(𝑎 = 𝐵𝑏 = ran 𝐴𝑥 = ⟨𝑎, 𝑏⟩) ↔ 𝑥 = ⟨𝐵, ran 𝐴⟩)
3311, 26, 323bitri 286 . . . . . . . . . . . . . 14 (⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥 = ⟨𝐵, ran 𝐴⟩)
3433anbi1i 733 . . . . . . . . . . . . 13 ((⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏) ↔ (𝑥 = ⟨𝐵, ran 𝐴⟩ ∧ 𝑥Cart𝑏))
3534exbii 1923 . . . . . . . . . . . 12 (∃𝑥(⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏) ↔ ∃𝑥(𝑥 = ⟨𝐵, ran 𝐴⟩ ∧ 𝑥Cart𝑏))
36 opex 5081 . . . . . . . . . . . . 13 𝐵, ran 𝐴⟩ ∈ V
37 breq1 4807 . . . . . . . . . . . . 13 (𝑥 = ⟨𝐵, ran 𝐴⟩ → (𝑥Cart𝑏 ↔ ⟨𝐵, ran 𝐴⟩Cart𝑏))
3836, 37ceqsexv 3382 . . . . . . . . . . . 12 (∃𝑥(𝑥 = ⟨𝐵, ran 𝐴⟩ ∧ 𝑥Cart𝑏) ↔ ⟨𝐵, ran 𝐴⟩Cart𝑏)
3935, 38bitri 264 . . . . . . . . . . 11 (∃𝑥(⟨𝐴, 𝐵⟩(2nd ⊗ (Range ∘ 1st ))𝑥𝑥Cart𝑏) ↔ ⟨𝐵, ran 𝐴⟩Cart𝑏)
407, 27, 9brcart 32345 . . . . . . . . . . 11 (⟨𝐵, ran 𝐴⟩Cart𝑏𝑏 = (𝐵 × ran 𝐴))
4110, 39, 403bitri 286 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏𝑏 = (𝐵 × ran 𝐴))
428, 41, 233anbi123i 1159 . . . . . . . . 9 ((⟨𝐴, 𝐵⟩1st 𝑎 ∧ ⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏𝑥 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝐴𝑏 = (𝐵 × ran 𝐴) ∧ 𝑥 = ⟨𝑎, 𝑏⟩))
435, 42bitri 264 . . . . . . . 8 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐴, 𝐵⟩1st 𝑎 ∧ ⟨𝐴, 𝐵⟩(Cart ∘ (2nd ⊗ (Range ∘ 1st )))𝑏) ↔ (𝑎 = 𝐴𝑏 = (𝐵 × ran 𝐴) ∧ 𝑥 = ⟨𝑎, 𝑏⟩))
44432exbii 1924 . . . . . . 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 7127 . . . . . . . 8 (𝐵 × ran 𝐴) ∈ V
46 opeq1 4553 . . . . . . . . 9 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
4746eqeq2d 2770 . . . . . . . 8 (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
48 opeq2 4554 . . . . . . . . 9 (𝑏 = (𝐵 × ran 𝐴) → ⟨𝐴, 𝑏⟩ = ⟨𝐴, (𝐵 × ran 𝐴)⟩)
4948eqeq2d 2770 . . . . . . . 8 (𝑏 = (𝐵 × ran 𝐴) → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩))
506, 45, 47, 49ceqsex2v 3385 . . . . . . 7 (∃𝑎𝑏(𝑎 = 𝐴𝑏 = (𝐵 × ran 𝐴) ∧ 𝑥 = ⟨𝑎, 𝑏⟩) ↔ 𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩)
514, 44, 503bitri 286 . . . . . 6 (⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩)
5251anbi1i 733 . . . . 5 ((⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥Cap𝐶) ↔ (𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶))
5352exbii 1923 . . . 4 (∃𝑥(⟨𝐴, 𝐵⟩(1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))𝑥𝑥Cap𝐶) ↔ ∃𝑥(𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶))
543, 53bitri 264 . . 3 (⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶 ↔ ∃𝑥(𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶))
55 opex 5081 . . . 4 𝐴, (𝐵 × ran 𝐴)⟩ ∈ V
56 breq1 4807 . . . 4 (𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ → (𝑥Cap𝐶 ↔ ⟨𝐴, (𝐵 × ran 𝐴)⟩Cap𝐶))
5755, 56ceqsexv 3382 . . 3 (∃𝑥(𝑥 = ⟨𝐴, (𝐵 × ran 𝐴)⟩ ∧ 𝑥Cap𝐶) ↔ ⟨𝐴, (𝐵 × ran 𝐴)⟩Cap𝐶)
586, 45, 2brcap 32353 . . 3 (⟨𝐴, (𝐵 × ran 𝐴)⟩Cap𝐶𝐶 = (𝐴 ∩ (𝐵 × ran 𝐴)))
5954, 57, 583bitri 286 . 2 (⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶𝐶 = (𝐴 ∩ (𝐵 × ran 𝐴)))
60 df-restrict 32284 . . 3 Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
6160breqi 4810 . 2 (⟨𝐴, 𝐵⟩Restrict𝐶 ↔ ⟨𝐴, 𝐵⟩(Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))𝐶)
62 dfres3 5556 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))
6362eqeq2i 2772 . 2 (𝐶 = (𝐴𝐵) ↔ 𝐶 = (𝐴 ∩ (𝐵 × ran 𝐴)))
6459, 61, 633bitr4i 292 1 (⟨𝐴, 𝐵⟩Restrict𝐶𝐶 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  cin 3714  cop 4327   class class class wbr 4804   × cxp 5264  ran crn 5267  cres 5268  ccom 5270  1st c1st 7331  2nd c2nd 7332  ctxp 32243  Cartccart 32254  Rangecrange 32257  Capccap 32260  Restrictcrestrict 32264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-symdif 3987  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-eprel 5179  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-1st 7333  df-2nd 7334  df-txp 32267  df-pprod 32268  df-image 32277  df-cart 32278  df-range 32281  df-cap 32283  df-restrict 32284
This theorem is referenced by:  dfrecs2  32363
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