Theorem f1eq3 6768

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 6682	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵))
2	1	anbi1d 643	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹) ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹)))
3		df-f1 6538	. 2 ⊢ (𝐹:𝐶–1-1→𝐴 ↔ (𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹))
4		df-f1 6538	. 2 ⊢ (𝐹:𝐶–1-1→𝐵 ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹))
5	2, 3, 4	3bitr4g 317	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ^◡ccnv 5654  Fun wfun 6527  ⟶wf 6529  –1-1→wf1 6530

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2752  df-ss 3916  df-f 6537  df-f1 6538

This theorem is used by:  f1oeq3  6807  f1eq123d  6809  tposf12  8249  brdom2g  8963  1sdom2dom  9224  pwfseq  10673  f1linds  22038  isusgrs  29616  usgrstrrepe  29695  usgrexilem  29900  tocycval  33548  diaf1oN  42003  f1cof1b  47965