Theorem dffun3f 42429

Description: Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.)

Hypotheses

Ref	Expression
dffun3f.1	|- F/_ x A
dffun3f.2	|- F/_ y A
dffun3f.3	|- F/_ z A

Assertion

Ref	Expression
dffun3f	|- ( Fun A <-> ( Rel A /\ A. x E. z A. y ( x A y -> y = z ) ) )

Distinct variable group:   x, y, z

Allowed substitution hints:   A( x, y, z)

Proof of Theorem dffun3f

Step	Hyp	Ref	Expression
1		dffun3f.1	. . 3 |- F/_ x A
2		dffun3f.2	. . 3 |- F/_ y A
3	1, 2	dffun6f 5902	. 2 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )
4		nfcv 2764	. . . . . 6 |- F/_ z x
5		dffun3f.3	. . . . . 6 |- F/_ z A
6		nfcv 2764	. . . . . 6 |- F/_ z y
7	4, 5, 6	nfbr 4699	. . . . 5 |- F/ z x A y
8	7	mo2 2479	. . . 4 |- ( E* y x A y <-> E. z A. y ( x A y -> y = z ) )
9	8	albii 1747	. . 3 |- ( A. x E* y x A y <-> A. x E. z A. y ( x A y -> y = z ) )
10	9	anbi2i 730	. 2 |- ( ( Rel A /\ A. x E* y x A y ) <-> ( Rel A /\ A. x E. z A. y ( x A y -> y = z ) ) )
11	3, 10	bitri 264	1 |- ( Fun A <-> ( Rel A /\ A. x E. z A. y ( x A y -> y = z ) ) )

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:  setrec2lem2  42441