Theorem feq3 5927

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 3590	. . 3 ⊢ (𝐴 = 𝐵 → (ran 𝐹 ⊆ 𝐴 ↔ ran 𝐹 ⊆ 𝐵))
2	1	anbi2d 736	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐴) ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐵)))
3		df-f 5794	. 2 ⊢ (𝐹:𝐶⟶𝐴 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐴))
4		df-f 5794	. 2 ⊢ (𝐹:𝐶⟶𝐵 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐵))
5	2, 3, 4	3bitr4g 302	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ⊆ wss 3540  ran crn 5029   Fn wfn 5785  ⟶wf 5786

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-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554  df-f 5794

This theorem is referenced by:  feq23  5928  feq3d  5931  fun2  5966  fconstg  5990  f1eq3  5996  mapvalg  7732  mapsn  7763  cantnff  8432  axdc4uz  12603  supcvg  14376  lmff  20863  txcn  21187  lmmbr  22809  iscmet3  22844  dvcnvrelem2  23530  itgsubstlem  23560  isumgra  25638  wlks  25841  wlkcompim  25848  wlkelwrd  25852  iseupa  26286  isgrpo  26529  vciOLD  26594  isvclem  26610  nmop0h  28028  sitgaddlemb  29531  sitmcl  29534  cvmliftlem15  30328  mtyf  30497  matunitlindflem1  32369  sdclem1  32503  k0004lem1  37259  mapsnd  38177  stoweidlem57  38744  hashfxnn0  40209  umgrislfupgr  40340  usgrislfuspgr  40406  1wlkv0  40851