Theorem dffun6f 4858

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/_xA
dffun6f.2	|- F/_yA

Assertion

Ref	Expression
dffun6f	|- ( Fun A <-> ( Rel A /\ A.xE*y xAy))

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 4855	. 2 |- ( Fun A <-> ( Rel A /\ A.wA.vA.u((wAv /\ wAu) -> v = u)))
2		nfcv 2175	. . . . . . 7 |- F/_yw
3		dffun6f.2	. . . . . . 7 |- F/_yA
4		nfcv 2175	. . . . . . 7 |- F/_yv
5	2, 3, 4	nfbr 3799	. . . . . 6 |- F/y wAv
6		nfv 1418	. . . . . 6 |- F/v wAy
7		breq2 3759	. . . . . 6 |- (v = y -> (wAv <-> wAy))
8	5, 6, 7	cbvmo 1937	. . . . 5 |- (E*v wAv <-> E*y wAy)
9	8	albii 1356	. . . 4 |- (A.wE*v wAv <-> A.wE*y wAy)
10		breq2 3759	. . . . . 6 |- (v = u -> (wAv <-> wAu))
11	10	mo4 1958	. . . . 5 |- (E*v wAv <-> A.vA.u((wAv /\ wAu) -> v = u))
12	11	albii 1356	. . . 4 |- (A.wE*v wAv <-> A.wA.vA.u((wAv /\ wAu) -> v = u))
13		nfcv 2175	. . . . . . 7 |- F/_xw
14		dffun6f.1	. . . . . . 7 |- F/_xA
15		nfcv 2175	. . . . . . 7 |- F/_xy
16	13, 14, 15	nfbr 3799	. . . . . 6 |- F/x wAy
17	16	nfmo 1917	. . . . 5 |- F/xE*y wAy
18		nfv 1418	. . . . 5 |- F/wE*y xAy
19		breq1 3758	. . . . . 6 |- (w = x -> (wAy <-> xAy))
20	19	mobidv 1933	. . . . 5 |- (w = x -> (E*y wAy <-> E*y xAy))
21	17, 18, 20	cbval 1634	. . . 4 |- (A.wE*y wAy <-> A.xE*y xAy)
22	9, 12, 21	3bitr3ri 200	. . 3 |- (A.xE*y xAy <-> A.wA.vA.u((wAv /\ wAu) -> v = u))
23	22	anbi2i 430	. 2 |- (( Rel A /\ A.xE*y xAy) <-> ( Rel A /\ A.wA.vA.u((wAv /\ wAu) -> v = u)))
24	1, 23	bitr4i 176	1 |- ( Fun A <-> ( Rel A /\ A.xE*y xAy))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240  E*wmo 1898   F/_wnfc 2162   class class class wbr 3755   Rel wrel 4293   Fun wfun 4839

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-cnv 4296  df-co 4297  df-fun 4847

This theorem is referenced by:  dffun6  4859  dffun4f  4861  funopab  4878