Theorem f1eq2 2777

Description: Equality theorem for one-to-one functions.

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 2749	. . 3 ⊢ (A = B → (F:A–→C ↔ F:B–→C))
2	1	anbi1d 469	. 2 ⊢ (A = B → ((F:A–→C ∧ Fun ◡F) ↔ (F:B–→C ∧ Fun ◡F)))
3		df-f1 2435	. 2 ⊢ (F:A–1-1→C ↔ (F:A–→C ∧ Fun ◡F))
4		df-f1 2435	. 2 ⊢ (F:B–1-1→C ↔ (F:B–→C ∧ Fun ◡F))
5	2, 3, 4	3bitr4g 428	1 ⊢ (A = B → (F:A–1-1→C ↔ F:B–1-1→C))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091  ^◡ccnv 2409  Fun wfun 2416  –→wf 2418  –1-1→wf1 2419

This theorem is referenced by:  f1oeq2 2796  brdomg 3281

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-cleq 1097  df-fn 2433  df-f 2434  df-f1 2435

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