Theorem feq3 6491

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 3992	. . 3 ⊢ (𝐴 = 𝐵 → (ran 𝐹 ⊆ 𝐴 ↔ ran 𝐹 ⊆ 𝐵))
2	1	anbi2d 630	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐴) ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐵)))
3		df-f 6353	. 2 ⊢ (𝐹:𝐶⟶𝐴 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐴))
4		df-f 6353	. 2 ⊢ (𝐹:𝐶⟶𝐵 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐵))
5	2, 3, 4	3bitr4g 316	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ⊆ wss 3935  ran crn 5550   Fn wfn 6344  ⟶wf 6345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-in 3942  df-ss 3951  df-f 6353

This theorem is referenced by:  feq23  6492  feq3d  6495  fun2  6535  fconstg  6560  f1eq3  6566  mapvalg  8410  mapsnd  8444  cantnff  9131  axdc4uz  13346  supcvg  15205  lmff  21903  txcn  22228  lmmbr  23855  iscmet3  23890  dvcnvrelem2  24609  itgsubstlem  24639  umgrislfupgr  26902  usgrislfuspgr  26963  wlkv0  27426  isgrpo  28268  vciOLD  28332  isvclem  28348  nmop0h  29762  sitgaddlemb  31601  sitmcl  31604  cvmliftlem15  32540  mtyf  32794  matunitlindflem1  34882  sdclem1  35012  k0004lem1  40490  stoweidlem57  42336  isomushgr  43985