Theorem dff1o4OLD 4645

Description: Alternate definition of one-to-one onto function.

Assertion

Ref	Expression
dff1o4OLD	|- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))

Proof of Theorem dff1o4OLD

Step	Hyp	Ref	Expression
1		3anass 862	. 2 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
2		dff1o2 4639	. 2 |- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))
3		df-fn 4009	. . . 4 |- (`'F Fn B <-> (Fun `'F /\ dom `' F = B))
4		df-rn 4005	. . . . . 6 |- ran F = dom `' F
5	4	eqeq1i 1891	. . . . 5 |- (ran F = B <-> dom `' F = B)
6	5	anbi2i 538	. . . 4 |- ((Fun `'F /\ ran F = B) <-> (Fun `'F /\ dom `' F = B))
7	3, 6	bitr4i 193	. . 3 |- (`'F Fn B <-> (Fun `'F /\ ran F = B))
8	7	anbi2i 538	. 2 |- ((F Fn A /\ `'F Fn B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
9	1, 2, 8	3bitr4i 200	1 |- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))

Syntax hints:   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298  `'ccnv 3985  dom cdm 3986  ran crn 3987  Fun wfun 3992   Fn wfn 3993  -1-1-onto->wf1o 3997

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-in 2603  df-ss 2605  df-rn 4005  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

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