Theorem f1eq2 5526

Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Assertion

Ref	Expression
f1eq2	⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶))

Proof of Theorem f1eq2

Step	Hyp	Ref	Expression
1		feq2 5456	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))
2	1	anbi1d 465	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐴⟶𝐶 ∧ Fun ◡𝐹) ↔ (𝐹:𝐵⟶𝐶 ∧ Fun ◡𝐹)))
3		df-f1 5322	. 2 ⊢ (𝐹:𝐴–1-1→𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ Fun ◡𝐹))
4		df-f1 5322	. 2 ⊢ (𝐹:𝐵–1-1→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ Fun ◡𝐹))
5	2, 3, 4	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ^◡ccnv 4717  Fun wfun 5311  ⟶wf 5313  –1-1→wf1 5314

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-fn 5320  df-f 5321  df-f1 5322

This theorem is referenced by:  f1oeq2  5560  f1eq123d  5563  f10d  5606  brdom2g  6894  brdomg  6895  ennnfonelemen  12987  ausgrusgrben  15960  dom1o  16314