Theorem dffun6f 5131

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 5128	. 2 |- ( Fun A <-> ( Rel A /\ A. w A. v A. u ( ( w A v /\ w A u ) -> v = u ) ) )
2		nfcv 2279	. . . . . . 7 |- F/_ y w
3		dffun6f.2	. . . . . . 7 |- F/_ y A
4		nfcv 2279	. . . . . . 7 |- F/_ y v
5	2, 3, 4	nfbr 3969	. . . . . 6 |- F/ y w A v
6		nfv 1508	. . . . . 6 |- F/ v w A y
7		breq2 3928	. . . . . 6 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvmo 2037	. . . . 5 |- ( E* v w A v <-> E* y w A y )
9	8	albii 1446	. . . 4 |- ( A. w E* v w A v <-> A. w E* y w A y )
10		breq2 3928	. . . . . 6 |- ( v = u -> ( w A v <-> w A u ) )
11	10	mo4 2058	. . . . 5 |- ( E* v w A v <-> A. v A. u ( ( w A v /\ w A u ) -> v = u ) )
12	11	albii 1446	. . . 4 |- ( A. w E* v w A v <-> A. w A. v A. u ( ( w A v /\ w A u ) -> v = u ) )
13		nfcv 2279	. . . . . . 7 |- F/_ x w
14		dffun6f.1	. . . . . . 7 |- F/_ x A
15		nfcv 2279	. . . . . . 7 |- F/_ x y
16	13, 14, 15	nfbr 3969	. . . . . 6 |- F/ x w A y
17	16	nfmo 2017	. . . . 5 |- F/ x E* y w A y
18		nfv 1508	. . . . 5 |- F/ w E* y x A y
19		breq1 3927	. . . . . 6 |- ( w = x -> ( w A y <-> x A y ) )
20	19	mobidv 2033	. . . . 5 |- ( w = x -> ( E* y w A y <-> E* y x A y ) )
21	17, 18, 20	cbval 1727	. . . 4 |- ( A. w E* y w A y <-> A. x E* y x A y )
22	9, 12, 21	3bitr3ri 210	. . 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 452	. 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 186	1 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329   E*wmo 1998   F/_wnfc 2266   class class class wbr 3924   Rel wrel 4539   Fun wfun 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-cnv 4542  df-co 4543  df-fun 5120

This theorem is referenced by:  dffun6  5132  dffun4f  5134  funopab  5153