Theorem dff1o2 5292

Description: Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 10-Feb-1997.) (Revised by set.mm contributors, 22-Oct-2011.)

Assertion

Ref	Expression
dff1o2	|- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))

Proof of Theorem dff1o2

Step	Hyp	Ref	Expression
1		df-f1o 4795	. 2 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
2		df-f1 4793	. . 3 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
3		df-fo 4794	. . 3 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
4	2, 3	anbi12i 678	. 2 |- ((F:A-1-1->B /\ F:A-onto->B) <-> ((F:A-->B /\ Fun `'F) /\ (F Fn A /\ ran F = B)))
5		ancom 437	. . . 4 |- ((F:A-->B /\ (Fun `'F /\ (F Fn A /\ ran F = B))) <-> ((Fun `'F /\ (F Fn A /\ ran F = B)) /\ F:A-->B))
6		3anass 938	. . . . . 6 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
7		an12 772	. . . . . 6 |- ((F Fn A /\ (Fun `'F /\ ran F = B)) <-> (Fun `'F /\ (F Fn A /\ ran F = B)))
8	6, 7	bitri 240	. . . . 5 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> (Fun `'F /\ (F Fn A /\ ran F = B)))
9	8	anbi1i 676	. . . 4 |- (((F Fn A /\ Fun `'F /\ ran F = B) /\ F:A-->B) <-> ((Fun `'F /\ (F Fn A /\ ran F = B)) /\ F:A-->B))
10	5, 9	bitr4i 243	. . 3 |- ((F:A-->B /\ (Fun `'F /\ (F Fn A /\ ran F = B))) <-> ((F Fn A /\ Fun `'F /\ ran F = B) /\ F:A-->B))
11		anass 630	. . 3 |- (((F:A-->B /\ Fun `'F) /\ (F Fn A /\ ran F = B)) <-> (F:A-->B /\ (Fun `'F /\ (F Fn A /\ ran F = B))))
12		eqimss 3324	. . . . . 6 |- (ran F = B -> ran F C_ B)
13		df-f 4792	. . . . . . 7 |- (F:A-->B <-> (F Fn A /\ ran F C_ B))
14	13	biimpri 197	. . . . . 6 |- ((F Fn A /\ ran F C_ B) -> F:A-->B)
15	12, 14	sylan2 460	. . . . 5 |- ((F Fn A /\ ran F = B) -> F:A-->B)
16	15	3adant2 974	. . . 4 |- ((F Fn A /\ Fun `'F /\ ran F = B) -> F:A-->B)
17	16	pm4.71i 613	. . 3 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> ((F Fn A /\ Fun `'F /\ ran F = B) /\ F:A-->B))
18	10, 11, 17	3bitr4i 268	. 2 |- (((F:A-->B /\ Fun `'F) /\ (F Fn A /\ ran F = B)) <-> (F Fn A /\ Fun `'F /\ ran F = B))
19	1, 4, 18	3bitri 262	1 |- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))

This proof depends on syntax axioms:   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   C_ wss 3258  `'ccnv 4772  ran crn 4774  Fun wfun 4776   Fn wfn 4777  -->wf 4778  -1-1->wf1 4779  -onto->wfo 4780  -1-1-onto->wf1o 4781

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

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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 used by:  dff1o3  5293  dff1o4  5295  f1orn  5297  fundmen  6044  enmap1  6075  enprmap  6083  sbthlem3  6206