Theorem feq2 5491

Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
feq2	⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))

Proof of Theorem feq2

Step	Hyp	Ref	Expression
1		fneq2 5444	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵))
2	1	anbi1d 465	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶)))
3		df-f 5355	. 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
4		df-f 5355	. 2 ⊢ (𝐹:𝐵⟶𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶))
5	2, 3, 4	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ⊆ wss 3210  ran crn 4749   Fn wfn 5346  ⟶wf 5347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-fn 5354  df-f 5355

This theorem is referenced by:  feq23  5493  feq2d  5495  feq2i  5501  f00  5558  f0dom0  5560  f1eq2  5568  fressnfv  5870  tfrcllemsucfn  6583  tfrcllemsucaccv  6584  tfrcllembxssdm  6586  tfrcllembfn  6587  tfrcllemaccex  6591  tfrcllemres  6592  tfrcldm  6593  tfrcl  6594  mapvalg  6891  map0g  6921  ac6sfi  7154  isomni  7426  ismkv  7443  iswomni  7455  isghm  13952