Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brcap Structured version   Visualization version   GIF version

Theorem brcap 34571
Description: Binary relation form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brcap.1 𝐴 ∈ V
brcap.2 𝐵 ∈ V
brcap.3 𝐶 ∈ V
Assertion
Ref Expression
brcap (⟨𝐴, 𝐵⟩Cap𝐶𝐶 = (𝐴𝐵))

Proof of Theorem brcap
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 5422 . 2 𝐴, 𝐵⟩ ∈ V
2 brcap.3 . 2 𝐶 ∈ V
3 df-cap 34501 . 2 Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∩ (2nd ∘ E )) ⊗ V)))
4 brcap.1 . . . 4 𝐴 ∈ V
5 brcap.2 . . . 4 𝐵 ∈ V
64, 5opelvv 5673 . . 3 𝐴, 𝐵⟩ ∈ (V × V)
7 brxp 5682 . . 3 (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
86, 2, 7mpbir2an 710 . 2 𝐴, 𝐵⟩((V × V) × V)𝐶
9 epel 5541 . . . . . . 7 (𝑥 E 𝑦𝑥𝑦)
10 vex 3448 . . . . . . . . 9 𝑦 ∈ V
1110, 1brcnv 5839 . . . . . . . 8 (𝑦1st𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩1st 𝑦)
124, 5br1steq 34401 . . . . . . . 8 (⟨𝐴, 𝐵⟩1st 𝑦𝑦 = 𝐴)
1311, 12bitri 275 . . . . . . 7 (𝑦1st𝐴, 𝐵⟩ ↔ 𝑦 = 𝐴)
149, 13anbi12ci 629 . . . . . 6 ((𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩) ↔ (𝑦 = 𝐴𝑥𝑦))
1514exbii 1851 . . . . 5 (∃𝑦(𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
16 vex 3448 . . . . . 6 𝑥 ∈ V
1716, 1brco 5827 . . . . 5 (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩))
184clel3 3614 . . . . 5 (𝑥𝐴 ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
1915, 17, 183bitr4i 303 . . . 4 (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥𝐴)
2010, 1brcnv 5839 . . . . . . . 8 (𝑦2nd𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩2nd 𝑦)
214, 5br2ndeq 34402 . . . . . . . 8 (⟨𝐴, 𝐵⟩2nd 𝑦𝑦 = 𝐵)
2220, 21bitri 275 . . . . . . 7 (𝑦2nd𝐴, 𝐵⟩ ↔ 𝑦 = 𝐵)
239, 22anbi12ci 629 . . . . . 6 ((𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩) ↔ (𝑦 = 𝐵𝑥𝑦))
2423exbii 1851 . . . . 5 (∃𝑦(𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
2516, 1brco 5827 . . . . 5 (𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩))
265clel3 3614 . . . . 5 (𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
2724, 25, 263bitr4i 303 . . . 4 (𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥𝐵)
2819, 27anbi12i 628 . . 3 ((𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ∧ 𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩) ↔ (𝑥𝐴𝑥𝐵))
29 brin 5158 . . 3 (𝑥((1st ∘ E ) ∩ (2nd ∘ E ))⟨𝐴, 𝐵⟩ ↔ (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ∧ 𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩))
30 elin 3927 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3128, 29, 303bitr4ri 304 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥((1st ∘ E ) ∩ (2nd ∘ E ))⟨𝐴, 𝐵⟩)
321, 2, 3, 8, 31brtxpsd3 34527 1 (⟨𝐴, 𝐵⟩Cap𝐶𝐶 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3444  cin 3910  cop 4593   class class class wbr 5106   E cep 5537   × cxp 5632  ccnv 5633  ccom 5638  1st c1st 7920  2nd c2nd 7921  Capccap 34478
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 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-symdif 4203  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-eprel 5538  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-1st 7922  df-2nd 7923  df-txp 34485  df-cap 34501
This theorem is referenced by:  brrestrict  34580
  Copyright terms: Public domain W3C validator