Theorem f1oeq2 5283

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 5255	. . 3 |- (A = B -> (F:A-1-1->C <-> F:B-1-1->C))
2		foeq2 5267	. . 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 4795	. 2 |- (F:A-1-1-onto->C <-> (F:A-1-1->C /\ F:A-onto->C))
5		df-f1o 4795	. 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 4779  -onto->wfo 4780  -1-1-onto->wf1o 4781

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 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795

This theorem is referenced by:  f1oeq23  5285  resin  5308  f1osng  5324  isoeq4  5486  bren  6031