Theorem feq2 5940

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 5894	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵))
2	1	anbi1d 737	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶)))
3		df-f 5808	. 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
4		df-f 5808	. 2 ⊢ (𝐹:𝐵⟶𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶))
5	2, 3, 4	3bitr4g 302	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ⊆ wss 3540  ran crn 5039   Fn wfn 5799  ⟶wf 5800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-cleq 2603  df-fn 5807  df-f 5808

This theorem is referenced by:  feq23  5942  feq2d  5944  feq2i  5950  f00  6000  f0dom0  6002  f1eq2  6010  fressnfv  6332  mapvalg  7754  map0g  7783  ac6sfi  8089  cofsmo  8974  axcc4dom  9146  ac6sg  9193  isghm  17483  pjdm2  19874  cmpcovf  21004  ulmval  23938  measval  29588  isrnmeas  29590  poseq  30994  soseq  30995  elno2  31051  noreson  31057  noxpsgn  31062  nodenselem6  31085  bj-finsumval0  32324  mbfresfi  32626  stoweidlem62  38955  hoidmvval0b  39480  vonioo  39573  vonicc  39576