Theorem dffun6f 4935

Description: Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
dffun6f.1	|- F/_ x A
dffun6f.2	|- F/_ y A

Assertion

Ref	Expression
dffun6f	|- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)

Proof of Theorem dffun6f

Dummy variables w v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun2 4932	. 2 |- ( Fun A <-> ( Rel A /\ A. w A. v A. u ( ( w A v /\ w A u ) -> v = u ) ) )
2		nfcv 2219	. . . . . . 7 |- F/_ y w
3		dffun6f.2	. . . . . . 7 |- F/_ y A
4		nfcv 2219	. . . . . . 7 |- F/_ y v
5	2, 3, 4	nfbr 3829	. . . . . 6 |- F/ y w A v
6		nfv 1461	. . . . . 6 |- F/ v w A y
7		breq2 3789	. . . . . 6 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvmo 1981	. . . . 5 |- ( E* v w A v <-> E* y w A y )
9	8	albii 1399	. . . 4 |- ( A. w E* v w A v <-> A. w E* y w A y )
10		breq2 3789	. . . . . 6 |- ( v = u -> ( w A v <-> w A u ) )
11	10	mo4 2002	. . . . 5 |- ( E* v w A v <-> A. v A. u ( ( w A v /\ w A u ) -> v = u ) )
12	11	albii 1399	. . . 4 |- ( A. w E* v w A v <-> A. w A. v A. u ( ( w A v /\ w A u ) -> v = u ) )
13		nfcv 2219	. . . . . . 7 |- F/_ x w
14		dffun6f.1	. . . . . . 7 |- F/_ x A
15		nfcv 2219	. . . . . . 7 |- F/_ x y
16	13, 14, 15	nfbr 3829	. . . . . 6 |- F/ x w A y
17	16	nfmo 1961	. . . . 5 |- F/ x E* y w A y
18		nfv 1461	. . . . 5 |- F/ w E* y x A y
19		breq1 3788	. . . . . 6 |- ( w = x -> ( w A y <-> x A y ) )
20	19	mobidv 1977	. . . . 5 |- ( w = x -> ( E* y w A y <-> E* y x A y ) )
21	17, 18, 20	cbval 1677	. . . 4 |- ( A. w E* y w A y <-> A. x E* y x A y )
22	9, 12, 21	3bitr3ri 209	. . 3 |- ( A. x E* y x A y <-> A. w A. v A. u ( ( w A v /\ w A u ) -> v = u ) )
23	22	anbi2i 444	. 2 |- ( ( Rel A /\ A. x E* y x A y ) <-> ( Rel A /\ A. w A. v A. u ( ( w A v /\ w A u ) -> v = u ) ) )
24	1, 23	bitr4i 185	1 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   E*wmo 1942   F/_wnfc 2206   class class class wbr 3785   Rel wrel 4368   Fun wfun 4916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-cnv 4371  df-co 4372  df-fun 4924

This theorem is referenced by:  dffun6  4936  dffun4f  4938  funopab  4955