Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  br1steqg Structured version   Visualization version   GIF version

Theorem br1steqg 30919
 Description: Uniqueness condition for binary relationship over the 1st relationship. (Contributed by Scott Fenton, 2-Jul-2020.)
Assertion
Ref Expression
br1steqg ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))

Proof of Theorem br1steqg
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 4340 . . . . . 6 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21breq1d 4593 . . . . 5 (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩1st 𝐶 ↔ ⟨𝐴, 𝑦⟩1st 𝐶))
3 eqeq2 2621 . . . . 5 (𝑥 = 𝐴 → (𝐶 = 𝑥𝐶 = 𝐴))
42, 3bibi12d 334 . . . 4 (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩1st 𝐶𝐶 = 𝑥) ↔ (⟨𝐴, 𝑦⟩1st 𝐶𝐶 = 𝐴)))
54imbi2d 329 . . 3 (𝑥 = 𝐴 → ((𝐶𝑋 → (⟨𝑥, 𝑦⟩1st 𝐶𝐶 = 𝑥)) ↔ (𝐶𝑋 → (⟨𝐴, 𝑦⟩1st 𝐶𝐶 = 𝐴))))
6 opeq2 4341 . . . . . 6 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
76breq1d 4593 . . . . 5 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩1st 𝐶 ↔ ⟨𝐴, 𝐵⟩1st 𝐶))
87bibi1d 332 . . . 4 (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩1st 𝐶𝐶 = 𝐴) ↔ (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)))
98imbi2d 329 . . 3 (𝑦 = 𝐵 → ((𝐶𝑋 → (⟨𝐴, 𝑦⟩1st 𝐶𝐶 = 𝐴)) ↔ (𝐶𝑋 → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))))
10 breq2 4587 . . . 4 (𝑧 = 𝐶 → (⟨𝑥, 𝑦⟩1st 𝑧 ↔ ⟨𝑥, 𝑦⟩1st 𝐶))
11 eqeq1 2614 . . . 4 (𝑧 = 𝐶 → (𝑧 = 𝑥𝐶 = 𝑥))
12 vex 3176 . . . . 5 𝑥 ∈ V
13 vex 3176 . . . . 5 𝑦 ∈ V
14 vex 3176 . . . . 5 𝑧 ∈ V
1512, 13, 14br1steq 30917 . . . 4 (⟨𝑥, 𝑦⟩1st 𝑧𝑧 = 𝑥)
1610, 11, 15vtoclbg 3240 . . 3 (𝐶𝑋 → (⟨𝑥, 𝑦⟩1st 𝐶𝐶 = 𝑥))
175, 9, 16vtocl2g 3243 . 2 ((𝐴𝑉𝐵𝑊) → (𝐶𝑋 → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)))
18173impia 1253 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  1st c1st 7057 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059 This theorem is referenced by:  fv1stcnv  30925
 Copyright terms: Public domain W3C validator