Theorem dffun6f 5558

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		dffun3 5555	. 2 |- ( Fun A <-> ( Rel A /\ A. w E. u A. v ( w A v -> v = u ) ) )
2		nfcv 2569	. . . . . . 7 |- F/_ y w
3		dffun6f.2	. . . . . . 7 |- F/_ y A
4		nfcv 2569	. . . . . . 7 |- F/_ y v
5	2, 3, 4	nfbr 4411	. . . . . 6 |- F/ y w A v
6		nfv 1755	. . . . . 6 |- F/ v w A y
7		breq2 4370	. . . . . 6 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvmo 2312	. . . . 5 |- ( E* v w A v <-> E* y w A y )
9	8	albii 1685	. . . 4 |- ( A. w E* v w A v <-> A. w E* y w A y )
10		mo2v 2283	. . . . 5 |- ( E* v w A v <-> E. u A. v ( w A v -> v = u ) )
11	10	albii 1685	. . . 4 |- ( A. w E* v w A v <-> A. w E. u A. v ( w A v -> v = u ) )
12		nfcv 2569	. . . . . . 7 |- F/_ x w
13		dffun6f.1	. . . . . . 7 |- F/_ x A
14		nfcv 2569	. . . . . . 7 |- F/_ x y
15	12, 13, 14	nfbr 4411	. . . . . 6 |- F/ x w A y
16	15	nfmo 2293	. . . . 5 |- F/ x E* y w A y
17		nfv 1755	. . . . 5 |- F/ w E* y x A y
18		breq1 4369	. . . . . 6 |- ( w = x -> ( w A y <-> x A y ) )
19	18	mobidv 2297	. . . . 5 |- ( w = x -> ( E* y w A y <-> E* y x A y ) )
20	16, 17, 19	cbval 2086	. . . 4 |- ( A. w E* y w A y <-> A. x E* y x A y )
21	9, 11, 20	3bitr3ri 279	. . 3 |- ( A. x E* y x A y <-> A. w E. u A. v ( w A v -> v = u ) )
22	21	anbi2i 698	. 2 |- ( ( Rel A /\ A. x E* y x A y ) <-> ( Rel A /\ A. w E. u A. v ( w A v -> v = u ) ) )
23	1, 22	bitr4i 255	1 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   A.wal 1435   E.wex 1657   E*wmo 2277   F/_wnfc 2556   class class class wbr 4366   Rel wrel 4801   Fun wfun 5538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-id 4711  df-cnv 4804  df-co 4805  df-fun 5546

This theorem is referenced by:  dffun6  5559  funopab  5577  funcnvmptOLD  28216  funcnvmpt  28217