New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  foeq3 GIF version

Theorem foeq3 5267
 Description: Equality theorem for onto functions. (Contributed by set.mm contributors, 1-Aug-1994.)
Assertion
Ref Expression
foeq3 (A = B → (F:ContoAF:ContoB))

Proof of Theorem foeq3
StepHypRef Expression
1 eqeq2 2362 . . 3 (A = B → (ran F = A ↔ ran F = B))
21anbi2d 684 . 2 (A = B → ((F Fn C ran F = A) ↔ (F Fn C ran F = B)))
3 df-fo 4793 . 2 (F:ContoA ↔ (F Fn C ran F = A))
4 df-fo 4793 . 2 (F:ContoB ↔ (F Fn C ran F = B))
52, 3, 43bitr4g 279 1 (A = B → (F:ContoAF:ContoB))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  ran crn 4773   Fn wfn 4776  –onto→wfo 4779 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-fo 4793 This theorem is referenced by:  f1oeq3  5283  resdif  5306  ffoss  5314
 Copyright terms: Public domain W3C validator