Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trcleq2lemRP Structured version   Visualization version   GIF version

Theorem trcleq2lemRP 40317
 Description: Equality implies bijection. (Contributed by RP, 5-May-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
trcleq2lemRP (𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

Proof of Theorem trcleq2lemRP
StepHypRef Expression
1 id 22 . . . 4 (𝐴 = 𝐵𝐴 = 𝐵)
21, 1coeq12d 5703 . . 3 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐵𝐵))
32, 1sseq12d 3951 . 2 (𝐴 = 𝐵 → ((𝐴𝐴) ⊆ 𝐴 ↔ (𝐵𝐵) ⊆ 𝐵))
43cleq2lem 40295 1 (𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ⊆ wss 3884   ∘ ccom 5527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901  df-br 5034  df-opab 5096  df-co 5532 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator