Theorem f1eq3 5420

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 5352	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵))
2	1	anbi1d 465	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹) ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹)))
3		df-f1 5223	. 2 ⊢ (𝐹:𝐶–1-1→𝐴 ↔ (𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹))
4		df-f1 5223	. 2 ⊢ (𝐹:𝐶–1-1→𝐵 ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹))
5	2, 3, 4	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ^◡ccnv 4627  Fun wfun 5212  ⟶wf 5214  –1-1→wf1 5215

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144  df-f 5222  df-f1 5223

This theorem is referenced by:  f1oeq3  5453  f1eq123d  5455  tposf12  6272  brdomg  6750