Theorem f1eq3 5114

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

Assertion

Ref	Expression
f1eq3	⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵))

Proof of Theorem f1eq3

Step	Hyp	Ref	Expression
1		feq3 5057	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵))
2	1	anbi1d 446	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹) ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹)))
3		df-f1 4932	. 2 ⊢ (𝐹:𝐶–1-1→𝐴 ↔ (𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹))
4		df-f1 4932	. 2 ⊢ (𝐹:𝐶–1-1→𝐵 ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹))
5	2, 3, 4	3bitr4g 216	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵))

Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1257  ^◡ccnv 4369  Fun wfun 4921  ⟶wf 4923  –1-1→wf1 4924

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-11 1411  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036

This theorem depends on definitions:  df-bi 114  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-in 2949  df-ss 2956  df-f 4931  df-f1 4932

This theorem is referenced by:  f1oeq3  5144  f1eq123d  5146  tposf12  5912  brdomg  6257