Theorem feq2 6065

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 6018	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵))
2	1	anbi1d 741	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶)))
3		df-f 5930	. 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
4		df-f 5930	. 2 ⊢ (𝐹:𝐵⟶𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶))
5	2, 3, 4	3bitr4g 303	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ⊆ wss 3607  ran crn 5144   Fn wfn 5921  ⟶wf 5922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-cleq 2644  df-fn 5929  df-f 5930

This theorem is referenced by:  feq23  6067  feq2d  6069  feq2i  6075  f00  6125  f0dom0  6127  f1eq2  6135  fressnfv  6467  mapvalg  7909  map0g  7939  ac6sfi  8245  cofsmo  9129  axcc4dom  9301  ac6sg  9348  isghm  17707  pjdm2  20103  cmpcovf  21242  ulmval  24179  measval  30389  isrnmeas  30391  poseq  31878  soseq  31879  elno2  31932  noreson  31938  bj-finsumval0  33277  mbfresfi  33586  stoweidlem62  40597  hoidmvval0b  41125  vonioo  41217  vonicc  41220