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Theorem feq3 5212
 Description: Equality theorem for functions. (Contributed by set.mm contributors, 1-Aug-1994.)
Assertion
Ref Expression
feq3 (A = B → (F:C–→AF:C–→B))

Proof of Theorem feq3
StepHypRef Expression
1 sseq2 3293 . . 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-f 4791 . 2 (F:C–→A ↔ (F Fn C ran F A))
4 df-f 4791 . 2 (F:C–→B ↔ (F Fn C ran F B))
52, 3, 43bitr4g 279 1 (A = B → (F:C–→AF:C–→B))
 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-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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-f 4791 This theorem is referenced by:  feq23  5213  fconstg  5251  f1eq3  5255  fsng  5433  fsn2  5434  mapex  6006  mapvalg  6009  mapsn  6026
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