Theorem f1oeq3 5284

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

Assertion

Ref	Expression
f1oeq3	|- (A = B -> (F:C-1-1-onto->A <-> F:C-1-1-onto->B))

Proof of Theorem f1oeq3

Step	Hyp	Ref	Expression
1		f1eq3 5256	. . 3 |- (A = B -> (F:C-1-1->A <-> F:C-1-1->B))
2		foeq3 5268	. . 3 |- (A = B -> (F:C-onto->A <-> F:C-onto->B))
3	1, 2	anbi12d 691	. 2 |- (A = B -> ((F:C-1-1->A /\ F:C-onto->A) <-> (F:C-1-1->B /\ F:C-onto->B)))
4		df-f1o 4795	. 2 |- (F:C-1-1-onto->A <-> (F:C-1-1->A /\ F:C-onto->A))
5		df-f1o 4795	. 2 |- (F:C-1-1-onto->B <-> (F:C-1-1->B /\ F:C-onto->B))
6	3, 4, 5	3bitr4g 279	1 |- (A = B -> (F:C-1-1-onto->A <-> F:C-1-1-onto->B))

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795

This theorem is referenced by:  f1oeq23  5285  resdif  5307  resin  5308  f1osng  5324  isoeq5  5487  isoini2  5499  swapres  5513  bren  6031  xpcomen  6053  xpassen  6058  enpw1pw  6076