Theorem dffun6f 5584

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 5581	. 2 |- ( Fun A <-> ( Rel A /\ A. w E. u A. v ( w A v -> v = u ) ) )
2		nfcv 2616	. . . . . . 7 |- F/_ y w
3		dffun6f.2	. . . . . . 7 |- F/_ y A
4		nfcv 2616	. . . . . . 7 |- F/_ y v
5	2, 3, 4	nfbr 4483	. . . . . 6 |- F/ y w A v
6		nfv 1712	. . . . . 6 |- F/ v w A y
7		breq2 4443	. . . . . 6 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvmo 2323	. . . . 5 |- ( E* v w A v <-> E* y w A y )
9	8	albii 1645	. . . 4 |- ( A. w E* v w A v <-> A. w E* y w A y )
10		mo2v 2291	. . . . 5 |- ( E* v w A v <-> E. u A. v ( w A v -> v = u ) )
11	10	albii 1645	. . . 4 |- ( A. w E* v w A v <-> A. w E. u A. v ( w A v -> v = u ) )
12		nfcv 2616	. . . . . . 7 |- F/_ x w
13		dffun6f.1	. . . . . . 7 |- F/_ x A
14		nfcv 2616	. . . . . . 7 |- F/_ x y
15	12, 13, 14	nfbr 4483	. . . . . 6 |- F/ x w A y
16	15	nfmo 2303	. . . . 5 |- F/ x E* y w A y
17		nfv 1712	. . . . 5 |- F/ w E* y x A y
18		breq1 4442	. . . . . 6 |- ( w = x -> ( w A y <-> x A y ) )
19	18	mobidv 2307	. . . . 5 |- ( w = x -> ( E* y w A y <-> E* y x A y ) )
20	16, 17, 19	cbval 2026	. . . 4 |- ( A. w E* y w A y <-> A. x E* y x A y )
21	9, 11, 20	3bitr3ri 276	. . 3 |- ( A. x E* y x A y <-> A. w E. u A. v ( w A v -> v = u ) )
22	21	anbi2i 692	. 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 252	1 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   A.wal 1396   E.wex 1617   E*wmo 2285   F/_wnfc 2602   class class class wbr 4439   Rel wrel 4993   Fun wfun 5564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-cnv 4996  df-co 4997  df-fun 5572

This theorem is referenced by:  dffun6  5585  funopab  5603  funcnvmptOLD  27739  funcnvmpt  27740