Theorem feq2 6667

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 6610	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵))
2	1	anbi1d 631	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶)))
3		df-f 6515	. 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
4		df-f 6515	. 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 1540   ⊆ wss 3914  ran crn 5639   Fn wfn 6506  ⟶wf 6507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721  df-fn 6514  df-f 6515

This theorem is referenced by:  feq23  6669  feq2d  6672  feq2i  6680  f00  6742  f0dom0  6744  f1eq2  6752  fressnfv  7132  poseq  8137  soseq  8138  mapvalg  8809  fsetexb  8837  map0g  8857  ac6sfi  9231  cofsmo  10222  axcc4dom  10394  ac6sg  10441  isghm  19147  isghmOLD  19148  pjdm2  21620  cmpcovf  23278  ulmval  26289  elno2  27566  noreson  27572  measval  34188  isrnmeas  34190  bj-finsumval0  37273  mbfresfi  37660  sn-isghm  42661  dfno2  43417  relpeq4  44937  stoweidlem62  46060  hoidmvval0b  46588  vonioo  46680  vonicc  46683  f102g  48840  f1mo  48841