New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  feq2 GIF version

Theorem feq2 5211
 Description: Equality theorem for functions. (Contributed by set.mm contributors, 1-Aug-1994.)
Assertion
Ref Expression
feq2 (A = B → (F:A–→CF:B–→C))

Proof of Theorem feq2
StepHypRef Expression
1 fneq2 5174 . . 3 (A = B → (F Fn AF Fn B))
21anbi1d 685 . 2 (A = B → ((F Fn A ran F C) ↔ (F Fn B ran F C)))
3 df-f 4791 . 2 (F:A–→C ↔ (F Fn A ran F C))
4 df-f 4791 . 2 (F:B–→C ↔ (F Fn B ran F C))
52, 3, 43bitr4g 279 1 (A = B → (F:A–→CF:B–→C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ⊆ wss 3257  ran crn 4773   Fn wfn 4776  –→wf 4777 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-fn 4790  df-f 4791 This theorem is referenced by:  feq23  5213  feq2d  5215  feq2i  5218  f00  5249  f1eq2  5254  fressnfv  5439  fconstfv  5456  mapex  6006  mapvalg  6009
 Copyright terms: Public domain W3C validator