Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  br1steq Structured version   Visualization version   GIF version

Theorem br1steq 31427
 Description: Uniqueness condition for binary relationship over the 1st relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
Hypotheses
Ref Expression
br1steq.1 𝐴 ∈ V
br1steq.2 𝐵 ∈ V
br1steq.3 𝐶 ∈ V
Assertion
Ref Expression
br1steq (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)

Proof of Theorem br1steq
StepHypRef Expression
1 br1steq.1 . . . 4 𝐴 ∈ V
2 br1steq.2 . . . 4 𝐵 ∈ V
31, 2op1st 7136 . . 3 (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
43eqeq1i 2626 . 2 ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶𝐴 = 𝐶)
5 fo1st 7148 . . . 4 1st :V–onto→V
6 fofn 6084 . . . 4 (1st :V–onto→V → 1st Fn V)
75, 6ax-mp 5 . . 3 1st Fn V
8 opex 4903 . . 3 𝐴, 𝐵⟩ ∈ V
9 fnbrfvb 6203 . . 3 ((1st Fn V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩1st 𝐶))
107, 8, 9mp2an 707 . 2 ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩1st 𝐶)
11 eqcom 2628 . 2 (𝐴 = 𝐶𝐶 = 𝐴)
124, 10, 113bitr3i 290 1 (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1480   ∈ wcel 1987  Vcvv 3190  ⟨cop 4161   class class class wbr 4623   Fn wfn 5852  –onto→wfo 5855  ‘cfv 5857  1st c1st 7126 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-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-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-fv 5865  df-1st 7128 This theorem is referenced by:  br1steqg  31429  dfdm5  31431  brtxp  31682  brpprod  31687  elfuns  31717  brimg  31739  brcup  31741  brcap  31742  brrestrict  31751
 Copyright terms: Public domain W3C validator