Theorem dffun6f 5390

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 5387	. 2 |- ( Fun A <-> ( Rel A /\ A. w A. v A. u ( ( w A v /\ w A u ) -> v = u ) ) )
2		nfcv 2392	. . . . . . 7 |- F/_ y w
3		dffun6f.2	. . . . . . 7 |- F/_ y A
4		nfcv 2392	. . . . . . 7 |- F/_ y v
5	2, 3, 4	nfbr 4177	. . . . . 6 |- F/ y w A v
6		nfv 1581	. . . . . 6 |- F/ v w A y
7		breq2 4134	. . . . . 6 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvmo 2126	. . . . 5 |- ( E* v w A v <-> E* y w A y )
9	8	albii 1523	. . . 4 |- ( A. w E* v w A v <-> A. w E* y w A y )
10		breq2 4134	. . . . . 6 |- ( v = u -> ( w A v <-> w A u ) )
11	10	mo4 2148	. . . . 5 |- ( E* v w A v <-> A. v A. u ( ( w A v /\ w A u ) -> v = u ) )
12	11	albii 1523	. . . 4 |- ( A. w E* v w A v <-> A. w A. v A. u ( ( w A v /\ w A u ) -> v = u ) )
13		nfcv 2392	. . . . . . 7 |- F/_ x w
14		dffun6f.1	. . . . . . 7 |- F/_ x A
15		nfcv 2392	. . . . . . 7 |- F/_ x y
16	13, 14, 15	nfbr 4177	. . . . . 6 |- F/ x w A y
17	16	nfmo 2106	. . . . 5 |- F/ x E* y w A y
18		nfv 1581	. . . . 5 |- F/ w E* y x A y
19		breq1 4133	. . . . . 6 |- ( w = x -> ( w A y <-> x A y ) )
20	19	mobidv 2122	. . . . 5 |- ( w = x -> ( E* y w A y <-> E* y x A y ) )
21	17, 18, 20	cbval 1807	. . . 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 461	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   E*wmo 2087   F/_wnfc 2379   class class class wbr 4130   Rel wrel 4779   Fun wfun 5371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-id 4438  df-cnv 4782  df-co 4783  df-fun 5379

This theorem is used by:  dffun6  5391  dffun4f  5393  funopab  5412