Theorem dffun6f 5516

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 5513	. 2 |- ( Fun A <-> ( Rel A /\ A. w E. u A. v ( w A v -> v = u ) ) )
2		nfcv 2610	. . . . . . 7 |- F/_ y w
3		dffun6f.2	. . . . . . 7 |- F/_ y A
4		nfcv 2610	. . . . . . 7 |- F/_ y v
5	2, 3, 4	nfbr 4420	. . . . . 6 |- F/ y w A v
6		nfv 1674	. . . . . 6 |- F/ v w A y
7		breq2 4380	. . . . . 6 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvmo 2303	. . . . 5 |- ( E* v w A v <-> E* y w A y )
9	8	albii 1611	. . . 4 |- ( A. w E* v w A v <-> A. w E* y w A y )
10		mo2v 2266	. . . . 5 |- ( E* v w A v <-> E. u A. v ( w A v -> v = u ) )
11	10	albii 1611	. . . 4 |- ( A. w E* v w A v <-> A. w E. u A. v ( w A v -> v = u ) )
12		nfcv 2610	. . . . . . 7 |- F/_ x w
13		dffun6f.1	. . . . . . 7 |- F/_ x A
14		nfcv 2610	. . . . . . 7 |- F/_ x y
15	12, 13, 14	nfbr 4420	. . . . . 6 |- F/ x w A y
16	15	nfmo 2279	. . . . 5 |- F/ x E* y w A y
17		nfv 1674	. . . . 5 |- F/ w E* y x A y
18		breq1 4379	. . . . . 6 |- ( w = x -> ( w A y <-> x A y ) )
19	18	mobidv 2283	. . . . 5 |- ( w = x -> ( E* y w A y <-> E* y x A y ) )
20	16, 17, 19	cbval 1970	. . . 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 694	. 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 369   A.wal 1368   E.wex 1587   E*wmo 2260   F/_wnfc 2596   class class class wbr 4376   Rel wrel 4929   Fun wfun 5496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rab 2801  df-v 3056  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-br 4377  df-opab 4435  df-id 4720  df-cnv 4932  df-co 4933  df-fun 5504

This theorem is referenced by:  dffun6  5517  funopab  5535  funcnvmptOLD  26106  funcnvmpt  26107