Theorem feq3 6650

Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
feq3	⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵))

Proof of Theorem feq3

Step	Hyp	Ref	Expression
1		sseq2 3962	. . 3 ⊢ (𝐴 = 𝐵 → (ran 𝐹 ⊆ 𝐴 ↔ ran 𝐹 ⊆ 𝐵))
2	1	anbi2d 631	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐴) ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐵)))
3		df-f 6504	. 2 ⊢ (𝐹:𝐶⟶𝐴 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐴))
4		df-f 6504	. 2 ⊢ (𝐹:𝐶⟶𝐵 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐵))
5	2, 3, 4	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ⊆ wss 3903  ran crn 5633   Fn wfn 6495  ⟶wf 6496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-ss 3920  df-f 6504

This theorem is referenced by:  feq23  6651  feq3d  6655  fun2  6705  fconstg  6729  f1eq3  6735  mapvalg  8785  mapsnd  8836  cantnff  9595  axdc4uz  13919  supcvg  15791  lmff  23257  txcn  23582  lmmbr  25226  iscmet3  25261  dvcnvrelem2  25991  itgsubstlem  26023  umgrislfupgr  29208  uspgriedgedg  29261  usgrislfuspgr  29272  wlkv0  29735  isgrpo  30585  vciOLD  30649  isvclem  30665  nmop0h  32079  sitgaddlemb  34526  sitmcl  34529  cvmliftlem15  35514  mtyf  35768  matunitlindflem1  37867  sdclem1  37994  k0004lem1  44503  relpeq5  45304  stoweidlem57  46415  f1ocof1ob  47441  isuspgrim0lem  48253  gricushgr  48277  uspgrlimlem4  48351  mof02  49198  mofsn2  49204  mofeu  49207  fdomne0  49209  f002  49213  fullthinc  49809  functermc  49867