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Theorem brrestrict 34921
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 𝐡 ∈ V
brrestrict.3 𝐢 ∈ V
Assertion
Ref Expression
brrestrict (⟨𝐴, 𝐡⟩Restrict𝐢 ↔ 𝐢 = (𝐴 β†Ύ 𝐡))

Proof of Theorem brrestrict
Dummy variables π‘Ž 𝑏 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 5465 . . . . 5 ⟨𝐴, 𝐡⟩ ∈ V
2 brrestrict.3 . . . . 5 𝐢 ∈ V
31, 2brco 5871 . . . 4 (⟨𝐴, 𝐡⟩(Cap ∘ (1st βŠ— (Cart ∘ (2nd βŠ— (Range ∘ 1st )))))𝐢 ↔ βˆƒπ‘₯(⟨𝐴, 𝐡⟩(1st βŠ— (Cart ∘ (2nd βŠ— (Range ∘ 1st ))))π‘₯ ∧ π‘₯Cap𝐢))
41brtxp2 34853 . . . . . . 7 (⟨𝐴, 𝐡⟩(1st βŠ— (Cart ∘ (2nd βŠ— (Range ∘ 1st ))))π‘₯ ↔ βˆƒπ‘Žβˆƒπ‘(π‘₯ = βŸ¨π‘Ž, π‘βŸ© ∧ ⟨𝐴, 𝐡⟩1st π‘Ž ∧ ⟨𝐴, 𝐡⟩(Cart ∘ (2nd βŠ— (Range ∘ 1st )))𝑏))
5 3anrot 1101 . . . . . . . . 9 ((π‘₯ = βŸ¨π‘Ž, π‘βŸ© ∧ ⟨𝐴, 𝐡⟩1st π‘Ž ∧ ⟨𝐴, 𝐡⟩(Cart ∘ (2nd βŠ— (Range ∘ 1st )))𝑏) ↔ (⟨𝐴, 𝐡⟩1st π‘Ž ∧ ⟨𝐴, 𝐡⟩(Cart ∘ (2nd βŠ— (Range ∘ 1st )))𝑏 ∧ π‘₯ = βŸ¨π‘Ž, π‘βŸ©))
6 brrestrict.1 . . . . . . . . . . 11 𝐴 ∈ V
7 brrestrict.2 . . . . . . . . . . 11 𝐡 ∈ V
86, 7br1steq 34742 . . . . . . . . . 10 (⟨𝐴, 𝐡⟩1st π‘Ž ↔ π‘Ž = 𝐴)
9 vex 3479 . . . . . . . . . . . 12 𝑏 ∈ V
101, 9brco 5871 . . . . . . . . . . 11 (⟨𝐴, 𝐡⟩(Cart ∘ (2nd βŠ— (Range ∘ 1st )))𝑏 ↔ βˆƒπ‘₯(⟨𝐴, 𝐡⟩(2nd βŠ— (Range ∘ 1st ))π‘₯ ∧ π‘₯Cart𝑏))
111brtxp2 34853 . . . . . . . . . . . . . . 15 (⟨𝐴, 𝐡⟩(2nd βŠ— (Range ∘ 1st ))π‘₯ ↔ βˆƒπ‘Žβˆƒπ‘(π‘₯ = βŸ¨π‘Ž, π‘βŸ© ∧ ⟨𝐴, 𝐡⟩2nd π‘Ž ∧ ⟨𝐴, 𝐡⟩(Range ∘ 1st )𝑏))
12 3anrot 1101 . . . . . . . . . . . . . . . . 17 ((π‘₯ = βŸ¨π‘Ž, π‘βŸ© ∧ ⟨𝐴, 𝐡⟩2nd π‘Ž ∧ ⟨𝐴, 𝐡⟩(Range ∘ 1st )𝑏) ↔ (⟨𝐴, 𝐡⟩2nd π‘Ž ∧ ⟨𝐴, 𝐡⟩(Range ∘ 1st )𝑏 ∧ π‘₯ = βŸ¨π‘Ž, π‘βŸ©))
136, 7br2ndeq 34743 . . . . . . . . . . . . . . . . . 18 (⟨𝐴, 𝐡⟩2nd π‘Ž ↔ π‘Ž = 𝐡)
141, 9brco 5871 . . . . . . . . . . . . . . . . . . 19 (⟨𝐴, 𝐡⟩(Range ∘ 1st )𝑏 ↔ βˆƒπ‘₯(⟨𝐴, 𝐡⟩1st π‘₯ ∧ π‘₯Range𝑏))
156, 7br1steq 34742 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝐴, 𝐡⟩1st π‘₯ ↔ π‘₯ = 𝐴)
1615anbi1i 625 . . . . . . . . . . . . . . . . . . . . 21 ((⟨𝐴, 𝐡⟩1st π‘₯ ∧ π‘₯Range𝑏) ↔ (π‘₯ = 𝐴 ∧ π‘₯Range𝑏))
1716exbii 1851 . . . . . . . . . . . . . . . . . . . 20 (βˆƒπ‘₯(⟨𝐴, 𝐡⟩1st π‘₯ ∧ π‘₯Range𝑏) ↔ βˆƒπ‘₯(π‘₯ = 𝐴 ∧ π‘₯Range𝑏))
18 breq1 5152 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ = 𝐴 β†’ (π‘₯Range𝑏 ↔ 𝐴Range𝑏))
196, 18ceqsexv 3526 . . . . . . . . . . . . . . . . . . . 20 (βˆƒπ‘₯(π‘₯ = 𝐴 ∧ π‘₯Range𝑏) ↔ 𝐴Range𝑏)
2017, 19bitri 275 . . . . . . . . . . . . . . . . . . 19 (βˆƒπ‘₯(⟨𝐴, 𝐡⟩1st π‘₯ ∧ π‘₯Range𝑏) ↔ 𝐴Range𝑏)
216, 9brrange 34906 . . . . . . . . . . . . . . . . . . 19 (𝐴Range𝑏 ↔ 𝑏 = ran 𝐴)
2214, 20, 213bitri 297 . . . . . . . . . . . . . . . . . 18 (⟨𝐴, 𝐡⟩(Range ∘ 1st )𝑏 ↔ 𝑏 = ran 𝐴)
23 biid 261 . . . . . . . . . . . . . . . . . 18 (π‘₯ = βŸ¨π‘Ž, π‘βŸ© ↔ π‘₯ = βŸ¨π‘Ž, π‘βŸ©)
2413, 22, 233anbi123i 1156 . . . . . . . . . . . . . . . . 17 ((⟨𝐴, 𝐡⟩2nd π‘Ž ∧ ⟨𝐴, 𝐡⟩(Range ∘ 1st )𝑏 ∧ π‘₯ = βŸ¨π‘Ž, π‘βŸ©) ↔ (π‘Ž = 𝐡 ∧ 𝑏 = ran 𝐴 ∧ π‘₯ = βŸ¨π‘Ž, π‘βŸ©))
2512, 24bitri 275 . . . . . . . . . . . . . . . 16 ((π‘₯ = βŸ¨π‘Ž, π‘βŸ© ∧ ⟨𝐴, 𝐡⟩2nd π‘Ž ∧ ⟨𝐴, 𝐡⟩(Range ∘ 1st )𝑏) ↔ (π‘Ž = 𝐡 ∧ 𝑏 = ran 𝐴 ∧ π‘₯ = βŸ¨π‘Ž, π‘βŸ©))
26252exbii 1852 . . . . . . . . . . . . . . 15 (βˆƒπ‘Žβˆƒπ‘(π‘₯ = βŸ¨π‘Ž, π‘βŸ© ∧ ⟨𝐴, 𝐡⟩2nd π‘Ž ∧ ⟨𝐴, 𝐡⟩(Range ∘ 1st )𝑏) ↔ βˆƒπ‘Žβˆƒπ‘(π‘Ž = 𝐡 ∧ 𝑏 = ran 𝐴 ∧ π‘₯ = βŸ¨π‘Ž, π‘βŸ©))
276rnex 7903 . . . . . . . . . . . . . . . 16 ran 𝐴 ∈ V
28 opeq1 4874 . . . . . . . . . . . . . . . . 17 (π‘Ž = 𝐡 β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨𝐡, π‘βŸ©)
2928eqeq2d 2744 . . . . . . . . . . . . . . . 16 (π‘Ž = 𝐡 β†’ (π‘₯ = βŸ¨π‘Ž, π‘βŸ© ↔ π‘₯ = ⟨𝐡, π‘βŸ©))
30 opeq2 4875 . . . . . . . . . . . . . . . . 17 (𝑏 = ran 𝐴 β†’ ⟨𝐡, π‘βŸ© = ⟨𝐡, ran 𝐴⟩)
3130eqeq2d 2744 . . . . . . . . . . . . . . . 16 (𝑏 = ran 𝐴 β†’ (π‘₯ = ⟨𝐡, π‘βŸ© ↔ π‘₯ = ⟨𝐡, ran 𝐴⟩))
327, 27, 29, 31ceqsex2v 3531 . . . . . . . . . . . . . . 15 (βˆƒπ‘Žβˆƒπ‘(π‘Ž = 𝐡 ∧ 𝑏 = ran 𝐴 ∧ π‘₯ = βŸ¨π‘Ž, π‘βŸ©) ↔ π‘₯ = ⟨𝐡, ran 𝐴⟩)
3311, 26, 323bitri 297 . . . . . . . . . . . . . 14 (⟨𝐴, 𝐡⟩(2nd βŠ— (Range ∘ 1st ))π‘₯ ↔ π‘₯ = ⟨𝐡, ran 𝐴⟩)
3433anbi1i 625 . . . . . . . . . . . . 13 ((⟨𝐴, 𝐡⟩(2nd βŠ— (Range ∘ 1st ))π‘₯ ∧ π‘₯Cart𝑏) ↔ (π‘₯ = ⟨𝐡, ran 𝐴⟩ ∧ π‘₯Cart𝑏))
3534exbii 1851 . . . . . . . . . . . 12 (βˆƒπ‘₯(⟨𝐴, 𝐡⟩(2nd βŠ— (Range ∘ 1st ))π‘₯ ∧ π‘₯Cart𝑏) ↔ βˆƒπ‘₯(π‘₯ = ⟨𝐡, ran 𝐴⟩ ∧ π‘₯Cart𝑏))
36 opex 5465 . . . . . . . . . . . . 13 ⟨𝐡, ran 𝐴⟩ ∈ V
37 breq1 5152 . . . . . . . . . . . . 13 (π‘₯ = ⟨𝐡, ran 𝐴⟩ β†’ (π‘₯Cart𝑏 ↔ ⟨𝐡, ran 𝐴⟩Cart𝑏))
3836, 37ceqsexv 3526 . . . . . . . . . . . 12 (βˆƒπ‘₯(π‘₯ = ⟨𝐡, ran 𝐴⟩ ∧ π‘₯Cart𝑏) ↔ ⟨𝐡, ran 𝐴⟩Cart𝑏)
3935, 38bitri 275 . . . . . . . . . . 11 (βˆƒπ‘₯(⟨𝐴, 𝐡⟩(2nd βŠ— (Range ∘ 1st ))π‘₯ ∧ π‘₯Cart𝑏) ↔ ⟨𝐡, ran 𝐴⟩Cart𝑏)
407, 27, 9brcart 34904 . . . . . . . . . . 11 (⟨𝐡, ran 𝐴⟩Cart𝑏 ↔ 𝑏 = (𝐡 Γ— ran 𝐴))
4110, 39, 403bitri 297 . . . . . . . . . 10 (⟨𝐴, 𝐡⟩(Cart ∘ (2nd βŠ— (Range ∘ 1st )))𝑏 ↔ 𝑏 = (𝐡 Γ— ran 𝐴))
428, 41, 233anbi123i 1156 . . . . . . . . 9 ((⟨𝐴, 𝐡⟩1st π‘Ž ∧ ⟨𝐴, 𝐡⟩(Cart ∘ (2nd βŠ— (Range ∘ 1st )))𝑏 ∧ π‘₯ = βŸ¨π‘Ž, π‘βŸ©) ↔ (π‘Ž = 𝐴 ∧ 𝑏 = (𝐡 Γ— ran 𝐴) ∧ π‘₯ = βŸ¨π‘Ž, π‘βŸ©))
435, 42bitri 275 . . . . . . . 8 ((π‘₯ = βŸ¨π‘Ž, π‘βŸ© ∧ ⟨𝐴, 𝐡⟩1st π‘Ž ∧ ⟨𝐴, 𝐡⟩(Cart ∘ (2nd βŠ— (Range ∘ 1st )))𝑏) ↔ (π‘Ž = 𝐴 ∧ 𝑏 = (𝐡 Γ— ran 𝐴) ∧ π‘₯ = βŸ¨π‘Ž, π‘βŸ©))
44432exbii 1852 . . . . . . 7 (βˆƒπ‘Žβˆƒπ‘(π‘₯ = βŸ¨π‘Ž, π‘βŸ© ∧ ⟨𝐴, 𝐡⟩1st π‘Ž ∧ ⟨𝐴, 𝐡⟩(Cart ∘ (2nd βŠ— (Range ∘ 1st )))𝑏) ↔ βˆƒπ‘Žβˆƒπ‘(π‘Ž = 𝐴 ∧ 𝑏 = (𝐡 Γ— ran 𝐴) ∧ π‘₯ = βŸ¨π‘Ž, π‘βŸ©))
457, 27xpex 7740 . . . . . . . 8 (𝐡 Γ— ran 𝐴) ∈ V
46 opeq1 4874 . . . . . . . . 9 (π‘Ž = 𝐴 β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨𝐴, π‘βŸ©)
4746eqeq2d 2744 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (π‘₯ = βŸ¨π‘Ž, π‘βŸ© ↔ π‘₯ = ⟨𝐴, π‘βŸ©))
48 opeq2 4875 . . . . . . . . 9 (𝑏 = (𝐡 Γ— ran 𝐴) β†’ ⟨𝐴, π‘βŸ© = ⟨𝐴, (𝐡 Γ— ran 𝐴)⟩)
4948eqeq2d 2744 . . . . . . . 8 (𝑏 = (𝐡 Γ— ran 𝐴) β†’ (π‘₯ = ⟨𝐴, π‘βŸ© ↔ π‘₯ = ⟨𝐴, (𝐡 Γ— ran 𝐴)⟩))
506, 45, 47, 49ceqsex2v 3531 . . . . . . 7 (βˆƒπ‘Žβˆƒπ‘(π‘Ž = 𝐴 ∧ 𝑏 = (𝐡 Γ— ran 𝐴) ∧ π‘₯ = βŸ¨π‘Ž, π‘βŸ©) ↔ π‘₯ = ⟨𝐴, (𝐡 Γ— ran 𝐴)⟩)
514, 44, 503bitri 297 . . . . . 6 (⟨𝐴, 𝐡⟩(1st βŠ— (Cart ∘ (2nd βŠ— (Range ∘ 1st ))))π‘₯ ↔ π‘₯ = ⟨𝐴, (𝐡 Γ— ran 𝐴)⟩)
5251anbi1i 625 . . . . 5 ((⟨𝐴, 𝐡⟩(1st βŠ— (Cart ∘ (2nd βŠ— (Range ∘ 1st ))))π‘₯ ∧ π‘₯Cap𝐢) ↔ (π‘₯ = ⟨𝐴, (𝐡 Γ— ran 𝐴)⟩ ∧ π‘₯Cap𝐢))
5352exbii 1851 . . . 4 (βˆƒπ‘₯(⟨𝐴, 𝐡⟩(1st βŠ— (Cart ∘ (2nd βŠ— (Range ∘ 1st ))))π‘₯ ∧ π‘₯Cap𝐢) ↔ βˆƒπ‘₯(π‘₯ = ⟨𝐴, (𝐡 Γ— ran 𝐴)⟩ ∧ π‘₯Cap𝐢))
543, 53bitri 275 . . 3 (⟨𝐴, 𝐡⟩(Cap ∘ (1st βŠ— (Cart ∘ (2nd βŠ— (Range ∘ 1st )))))𝐢 ↔ βˆƒπ‘₯(π‘₯ = ⟨𝐴, (𝐡 Γ— ran 𝐴)⟩ ∧ π‘₯Cap𝐢))
55 opex 5465 . . . 4 ⟨𝐴, (𝐡 Γ— ran 𝐴)⟩ ∈ V
56 breq1 5152 . . . 4 (π‘₯ = ⟨𝐴, (𝐡 Γ— ran 𝐴)⟩ β†’ (π‘₯Cap𝐢 ↔ ⟨𝐴, (𝐡 Γ— ran 𝐴)⟩Cap𝐢))
5755, 56ceqsexv 3526 . . 3 (βˆƒπ‘₯(π‘₯ = ⟨𝐴, (𝐡 Γ— ran 𝐴)⟩ ∧ π‘₯Cap𝐢) ↔ ⟨𝐴, (𝐡 Γ— ran 𝐴)⟩Cap𝐢)
586, 45, 2brcap 34912 . . 3 (⟨𝐴, (𝐡 Γ— ran 𝐴)⟩Cap𝐢 ↔ 𝐢 = (𝐴 ∩ (𝐡 Γ— ran 𝐴)))
5954, 57, 583bitri 297 . 2 (⟨𝐴, 𝐡⟩(Cap ∘ (1st βŠ— (Cart ∘ (2nd βŠ— (Range ∘ 1st )))))𝐢 ↔ 𝐢 = (𝐴 ∩ (𝐡 Γ— ran 𝐴)))
60 df-restrict 34843 . . 3 Restrict = (Cap ∘ (1st βŠ— (Cart ∘ (2nd βŠ— (Range ∘ 1st )))))
6160breqi 5155 . 2 (⟨𝐴, 𝐡⟩Restrict𝐢 ↔ ⟨𝐴, 𝐡⟩(Cap ∘ (1st βŠ— (Cart ∘ (2nd βŠ— (Range ∘ 1st )))))𝐢)
62 dfres3 5987 . . 3 (𝐴 β†Ύ 𝐡) = (𝐴 ∩ (𝐡 Γ— ran 𝐴))
6362eqeq2i 2746 . 2 (𝐢 = (𝐴 β†Ύ 𝐡) ↔ 𝐢 = (𝐴 ∩ (𝐡 Γ— ran 𝐴)))
6459, 61, 633bitr4i 303 1 (⟨𝐴, 𝐡⟩Restrict𝐢 ↔ 𝐢 = (𝐴 β†Ύ 𝐡))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948  βŸ¨cop 4635   class class class wbr 5149   Γ— cxp 5675  ran crn 5678   β†Ύ cres 5679   ∘ ccom 5681  1st c1st 7973  2nd c2nd 7974   βŠ— ctxp 34802  Cartccart 34813  Rangecrange 34816  Capccap 34819  Restrictcrestrict 34823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-symdif 4243  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975  df-2nd 7976  df-txp 34826  df-pprod 34827  df-image 34836  df-cart 34837  df-range 34840  df-cap 34842  df-restrict 34843
This theorem is referenced by:  dfrecs2  34922
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