Theorem dffun6f 5370

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 5367	. 2 |- ( Fun A <-> ( Rel A /\ A. w A. v A. u ( ( w A v /\ w A u ) -> v = u ) ) )
2		nfcv 2386	. . . . . . 7 |- F/_ y w
3		dffun6f.2	. . . . . . 7 |- F/_ y A
4		nfcv 2386	. . . . . . 7 |- F/_ y v
5	2, 3, 4	nfbr 4161	. . . . . 6 |- F/ y w A v
6		nfv 1577	. . . . . 6 |- F/ v w A y
7		breq2 4118	. . . . . 6 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvmo 2122	. . . . 5 |- ( E* v w A v <-> E* y w A y )
9	8	albii 1519	. . . 4 |- ( A. w E* v w A v <-> A. w E* y w A y )
10		breq2 4118	. . . . . 6 |- ( v = u -> ( w A v <-> w A u ) )
11	10	mo4 2144	. . . . 5 |- ( E* v w A v <-> A. v A. u ( ( w A v /\ w A u ) -> v = u ) )
12	11	albii 1519	. . . 4 |- ( A. w E* v w A v <-> A. w A. v A. u ( ( w A v /\ w A u ) -> v = u ) )
13		nfcv 2386	. . . . . . 7 |- F/_ x w
14		dffun6f.1	. . . . . . 7 |- F/_ x A
15		nfcv 2386	. . . . . . 7 |- F/_ x y
16	13, 14, 15	nfbr 4161	. . . . . 6 |- F/ x w A y
17	16	nfmo 2102	. . . . 5 |- F/ x E* y w A y
18		nfv 1577	. . . . 5 |- F/ w E* y x A y
19		breq1 4117	. . . . . 6 |- ( w = x -> ( w A y <-> x A y ) )
20	19	mobidv 2118	. . . . 5 |- ( w = x -> ( E* y w A y <-> E* y x A y ) )
21	17, 18, 20	cbval 1803	. . . 4 |- ( A. w E* y w A y <-> A. x E* y x A y )
22	9, 12, 21	3bitr3ri 211	. . 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 457	. 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 187	1 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1396   E*wmo 2083   F/_wnfc 2373   class class class wbr 4114   Rel wrel 4759   Fun wfun 5351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-cnv 4762  df-co 4763  df-fun 5359

This theorem is referenced by:  dffun6  5371  dffun4f  5373  funopab  5392