Theorem feq2 6729

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 6671	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵))
2	1	anbi1d 630	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶)))
3		df-f 6577	. 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
4		df-f 6577	. 2 ⊢ (𝐹:𝐵⟶𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶))
5	2, 3, 4	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ⊆ wss 3976  ran crn 5701   Fn wfn 6568  ⟶wf 6569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732  df-fn 6576  df-f 6577

This theorem is referenced by:  feq23  6731  feq2d  6733  feq2i  6739  f00  6803  f0dom0  6805  f1eq2  6813  fressnfv  7194  poseq  8199  soseq  8200  mapvalg  8894  fsetexb  8922  map0g  8942  ac6sfi  9348  cofsmo  10338  axcc4dom  10510  ac6sg  10557  isghm  19255  isghmOLD  19256  pjdm2  21754  cmpcovf  23420  ulmval  26441  elno2  27717  noreson  27723  measval  34162  isrnmeas  34164  bj-finsumval0  37251  mbfresfi  37626  sn-isghm  42628  dfno2  43390  stoweidlem62  45983  hoidmvval0b  46511  vonioo  46603  vonicc  46606  f102g  48565  f1mo  48566