Theorem dffun6f 5902

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 5899	. 2 |- ( Fun A <-> ( Rel A /\ A. w E. u A. v ( w A v -> v = u ) ) )
2		nfcv 2764	. . . . . . 7 |- F/_ y w
3		dffun6f.2	. . . . . . 7 |- F/_ y A
4		nfcv 2764	. . . . . . 7 |- F/_ y v
5	2, 3, 4	nfbr 4699	. . . . . 6 |- F/ y w A v
6		nfv 1843	. . . . . 6 |- F/ v w A y
7		breq2 4657	. . . . . 6 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvmo 2506	. . . . 5 |- ( E* v w A v <-> E* y w A y )
9	8	albii 1747	. . . 4 |- ( A. w E* v w A v <-> A. w E* y w A y )
10		mo2v 2477	. . . . 5 |- ( E* v w A v <-> E. u A. v ( w A v -> v = u ) )
11	10	albii 1747	. . . 4 |- ( A. w E* v w A v <-> A. w E. u A. v ( w A v -> v = u ) )
12		nfcv 2764	. . . . . . 7 |- F/_ x w
13		dffun6f.1	. . . . . . 7 |- F/_ x A
14		nfcv 2764	. . . . . . 7 |- F/_ x y
15	12, 13, 14	nfbr 4699	. . . . . 6 |- F/ x w A y
16	15	nfmo 2487	. . . . 5 |- F/ x E* y w A y
17		nfv 1843	. . . . 5 |- F/ w E* y x A y
18		breq1 4656	. . . . . 6 |- ( w = x -> ( w A y <-> x A y ) )
19	18	mobidv 2491	. . . . 5 |- ( w = x -> ( E* y w A y <-> E* y x A y ) )
20	16, 17, 19	cbval 2271	. . . 4 |- ( A. w E* y w A y <-> A. x E* y x A y )
21	9, 11, 20	3bitr3ri 291	. . 3 |- ( A. x E* y x A y <-> A. w E. u A. v ( w A v -> v = u ) )
22	21	anbi2i 730	. 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 267	1 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   E*wmo 2471   F/_wnfc 2751   class class class wbr 4653   Rel wrel 5119   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  dffun6  5903  funopab  5923  funcnvmptOLD  29467  funcnvmpt  29468  dffun3f  42429