Theorem dffunmof 2678

Description: Function definition requiring only that x and y not be "free" in A (but not necessarily absent from it), using "at most one" notation.

Hypotheses

Ref	Expression
dffunmof.1	|- (z e. A -> A.x z e. A)
dffunmof.2	|- (z e. A -> A.y z e. A)

Assertion

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

Distinct variable group(s):   x,y,z   z,A

Proof of Theorem dffunmof

Step	Hyp	Ref	Expression
1		dffun3 2675	. 2 |- (Fun A <-> (Rel A /\ A.wE.uA.v(wAv -> v = u)))
2		ax-17 925	. . . . . . 7 |- (z e. w -> A.y z e. w)
3		dffunmof.2	. . . . . . 7 |- (z e. A -> A.y z e. A)
4		ax-17 925	. . . . . . 7 |- (z e. v -> A.y z e. v)
5	2, 3, 4	hbbr 2095	. . . . . 6 |- (wAv -> A.y wAv)
6		ax-17 925	. . . . . 6 |- (wAy -> A.v wAy)
7		breq2 2066	. . . . . 6 |- (v = y -> (wAv <-> wAy))
8	5, 6, 7	cbvmo 1034	. . . . 5 |- (E*v wAv <-> E*y wAy)
9	8	bial 695	. . . 4 |- (A.wE*v wAv <-> A.wE*y wAy)
10		ax-17 925	. . . . . 6 |- (wAv -> A.u wAv)
11	10	mo2 1026	. . . . 5 |- (E*v wAv <-> E.uA.v(wAv -> v = u))
12	11	bial 695	. . . 4 |- (A.wE*v wAv <-> A.wE.uA.v(wAv -> v = u))
13		ax-17 925	. . . . . . 7 |- (z e. w -> A.x z e. w)
14		dffunmof.1	. . . . . . 7 |- (z e. A -> A.x z e. A)
15		ax-17 925	. . . . . . 7 |- (z e. y -> A.x z e. y)
16	13, 14, 15	hbbr 2095	. . . . . 6 |- (wAy -> A.x wAy)
17	16	hbmo 1033	. . . . 5 |- (E*y wAy -> A.xE*y wAy)
18		ax-17 925	. . . . 5 |- (E*y xAy -> A.wE*y xAy)
19		ax-17 925	. . . . . 6 |- (w = x -> A.y w = x)
20		breq1 2065	. . . . . 6 |- (w = x -> (wAy <-> xAy))
21	19, 20	bimod 1030	. . . . 5 |- (w = x -> (E*y wAy <-> E*y xAy))
22	17, 18, 21	cbval 848	. . . 4 |- (A.wE*y wAy <-> A.xE*y xAy)
23	9, 12, 22	3bitr3r 157	. . 3 |- (A.xE*y xAy <-> A.wE.uA.v(wAv -> v = u))
24	23	anbi2i 367	. 2 |- ((Rel A /\ A.xE*y xAy) <-> (Rel A /\ A.wE.uA.v(wAv -> v = u)))
25	1, 24	bitr4 154	1 |- (Fun A <-> (Rel A /\ A.xE*y xAy))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803  E*wmo 1008   e. wcel 1092   class class class wbr 2054  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  dffunmo 2679  funopab 2694

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-cnv 2426  df-co 2427  df-fun 2432

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