Theorem f1oeq2 5282

Description: Equality theorem for one-to-one onto functions. (Contributed by set.mm contributors, 10-Feb-1997.)

Assertion

Ref	Expression
f1oeq2	⊢ (A = B → (F:A–1-1-onto→C ↔ F:B–1-1-onto→C))

Proof of Theorem f1oeq2

Step	Hyp	Ref	Expression
1		f1eq2 5254	. . 3 ⊢ (A = B → (F:A–1-1→C ↔ F:B–1-1→C))
2		foeq2 5266	. . 3 ⊢ (A = B → (F:A–onto→C ↔ F:B–onto→C))
3	1, 2	anbi12d 691	. 2 ⊢ (A = B → ((F:A–1-1→C ∧ F:A–onto→C) ↔ (F:B–1-1→C ∧ F:B–onto→C)))
4		df-f1o 4794	. 2 ⊢ (F:A–1-1-onto→C ↔ (F:A–1-1→C ∧ F:A–onto→C))
5		df-f1o 4794	. 2 ⊢ (F:B–1-1-onto→C ↔ (F:B–1-1→C ∧ F:B–onto→C))
6	3, 4, 5	3bitr4g 279	1 ⊢ (A = B → (F:A–1-1-onto→C ↔ F:B–1-1-onto→C))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  –1-1→wf1 4778  –onto→wfo 4779  –1-1-onto→wf1o 4780

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794

This theorem is referenced by:  f1oeq23  5284  resin  5307  f1osng  5323  isoeq4  5485  bren  6030