Theorem f1eq3 6805

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

Assertion

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

Proof of Theorem f1eq3

Step	Hyp	Ref	Expression
1		feq3 6719	. . 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 6571	. 2 ⊢ (𝐹:𝐶⟶1-1𝐴 ↔ (𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹))
4		df-f1 6571	. 2 ⊢ (𝐹:𝐶⟶1-1𝐵 ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹))
5	2, 3, 4	3bitr4g 317	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶1-1𝐴 ↔ 𝐹:𝐶⟶1-1𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ^◡ccnv 5657  Fun wfun 6560  ⟶wf 6562  ⟶1-1wf1 6563

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 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-ss 3901  df-f 6570  df-f1 6571

This theorem is used by:  f1oeq3  6844  f1eq123d  6846  tposf12  8306  brdom2g  9022  1sdom2dom  9283  pwfseq  10788  f1linds  22186  isusgrs  29794  usgrstrrepe  29873  usgrexilem  30078  tocycval  33768  diaf1oN  42453  f1cof1b  48420