Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  feq3 GIF version

Theorem feq3 5257
 Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
feq3 (𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))

Proof of Theorem feq3
StepHypRef Expression
1 sseq2 3121 . . 3 (𝐴 = 𝐵 → (ran 𝐹𝐴 ↔ ran 𝐹𝐵))
21anbi2d 459 . 2 (𝐴 = 𝐵 → ((𝐹 Fn 𝐶 ∧ ran 𝐹𝐴) ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹𝐵)))
3 df-f 5127 . 2 (𝐹:𝐶𝐴 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹𝐴))
4 df-f 5127 . 2 (𝐹:𝐶𝐵 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹𝐵))
52, 3, 43bitr4g 222 1 (𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ⊆ wss 3071  ran crn 4540   Fn wfn 5118  ⟶wf 5119 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-f 5127 This theorem is referenced by:  feq23  5258  feq3d  5261  feq123d  5263  fun2  5296  fconstg  5319  f1eq3  5325  fsng  5593  fsn2  5594  fsnunf  5620  mapvalg  6552  mapsn  6584  lmff  12428  txcn  12454
 Copyright terms: Public domain W3C validator