Theorem dffun6f 5614

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 5611	. 2 |- ( Fun A <-> ( Rel A /\ A. w E. u A. v ( w A v -> v = u ) ) )
2		nfcv 2602	. . . . . . 7 |- F/_ y w
3		dffun6f.2	. . . . . . 7 |- F/_ y A
4		nfcv 2602	. . . . . . 7 |- F/_ y v
5	2, 3, 4	nfbr 4460	. . . . . 6 |- F/ y w A v
6		nfv 1771	. . . . . 6 |- F/ v w A y
7		breq2 4419	. . . . . 6 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvmo 2345	. . . . 5 |- ( E* v w A v <-> E* y w A y )
9	8	albii 1701	. . . 4 |- ( A. w E* v w A v <-> A. w E* y w A y )
10		mo2v 2316	. . . . 5 |- ( E* v w A v <-> E. u A. v ( w A v -> v = u ) )
11	10	albii 1701	. . . 4 |- ( A. w E* v w A v <-> A. w E. u A. v ( w A v -> v = u ) )
12		nfcv 2602	. . . . . . 7 |- F/_ x w
13		dffun6f.1	. . . . . . 7 |- F/_ x A
14		nfcv 2602	. . . . . . 7 |- F/_ x y
15	12, 13, 14	nfbr 4460	. . . . . 6 |- F/ x w A y
16	15	nfmo 2326	. . . . 5 |- F/ x E* y w A y
17		nfv 1771	. . . . 5 |- F/ w E* y x A y
18		breq1 4418	. . . . . 6 |- ( w = x -> ( w A y <-> x A y ) )
19	18	mobidv 2330	. . . . 5 |- ( w = x -> ( E* y w A y <-> E* y x A y ) )
20	16, 17, 19	cbval 2124	. . . 4 |- ( A. w E* y w A y <-> A. x E* y x A y )
21	9, 11, 20	3bitr3ri 284	. . 3 |- ( A. x E* y x A y <-> A. w E. u A. v ( w A v -> v = u ) )
22	21	anbi2i 705	. 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 260	1 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   A.wal 1452   E.wex 1673   E*wmo 2310   F/_wnfc 2589   class class class wbr 4415   Rel wrel 4857   Fun wfun 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-br 4416  df-opab 4475  df-id 4767  df-cnv 4860  df-co 4861  df-fun 5602

This theorem is referenced by:  dffun6  5615  funopab  5633  funcnvmptOLD  28318  funcnvmpt  28319