Theorem feq3 5365

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 3194	. . 3 ⊢ (𝐴 = 𝐵 → (ran 𝐹 ⊆ 𝐴 ↔ ran 𝐹 ⊆ 𝐵))
2	1	anbi2d 464	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐴) ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐵)))
3		df-f 5235	. 2 ⊢ (𝐹:𝐶⟶𝐴 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐴))
4		df-f 5235	. 2 ⊢ (𝐹:𝐶⟶𝐵 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐵))
5	2, 3, 4	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ⊆ wss 3144  ran crn 4642   Fn wfn 5226  ⟶wf 5227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157  df-f 5235

This theorem is referenced by:  feq23  5366  feq3d  5369  feq123d  5371  fun2  5404  fconstg  5427  f1eq3  5433  fsng  5705  fsn2  5706  fsnunf  5732  mapvalg  6676  mapsn  6708  lmff  14133  txcn  14159