Theorem feq2 6686

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 6629	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵))
2	1	anbi1d 643	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶)))
3		df-f 6541	. 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
4		df-f 6541	. 2 ⊢ (𝐹:𝐵⟶𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶))
5	2, 3, 4	3bitr4g 317	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ⊆ wss 3899  ran crn 5652   Fn wfn 6532  ⟶wf 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2753  df-fn 6540  df-f 6541

This theorem is used by:  feq23  6688  feq2d  6691  feq2i  6699  f00  6762  f0dom0  6764  f1eq2  6772  fressnfv  7162  poseq  8168  soseq  8169  mapvalg  8849  fsetexb  8879  map0g  8905  ac6sfi  9268  cofsmo  10340  axcc4dom  10512  ac6sg  10559  isghm  19423  pjdm2  22010  cmpcovf  23702  ulmval  26700  elno2  28004  noreson  28010  measval  34824  isrnmeas  34826  bj-finsumval0  38186  mbfresfi  38564  sn-isghm  43664  dfno2  44413  relpeq4  45915  stoweidlem62  47041  hoidmvval0b  47569  vonioo  47661  vonicc  47664  f102g  49931  f1mo  49932