Theorem f1eq2 5255

Description: Equality theorem for one-to-one functions. (Contributed by set.mm contributors, 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 5212	. . 3 ⊢ (A = B → (F:A–→C ↔ F:B–→C))
2	1	anbi1d 685	. 2 ⊢ (A = B → ((F:A–→C ∧ Fun ◡F) ↔ (F:B–→C ∧ Fun ◡F)))
3		df-f1 4793	. 2 ⊢ (F:A–1-1→C ↔ (F:A–→C ∧ Fun ◡F))
4		df-f1 4793	. 2 ⊢ (F:B–1-1→C ↔ (F:B–→C ∧ Fun ◡F))
5	2, 3, 4	3bitr4g 279	1 ⊢ (A = B → (F:A–1-1→C ↔ F:B–1-1→C))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  ^◡ccnv 4772  Fun wfun 4776  –→wf 4778  –1-1→wf1 4779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-fn 4791  df-f 4792  df-f1 4793

This theorem is used by:  f1oeq2  5283  dflec3  6222  nclenc  6223