Theorem dffunmof 3546

Description: Definition of function, using bound-variable hypotheses instead of distinct variable conditions.

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 groups:   x,y,z   z,A

Proof of Theorem dffunmof

Step	Hyp	Ref	Expression
1		dffun3 3543	. 2 |- (Fun A <-> (Rel A /\ A.wE.uA.v(wAv -> v = u)))
2		ax-17 975	. . . . . . 7 |- (z e. w -> A.y z e. w)
3		dffunmof.2	. . . . . . 7 |- (z e. A -> A.y z e. A)
4		ax-17 975	. . . . . . 7 |- (z e. v -> A.y z e. v)
5	2, 3, 4	hbbr 2673	. . . . . 6 |- (wAv -> A.y wAv)
6		ax-17 975	. . . . . 6 |- (wAy -> A.v wAy)
7		breq2 2638	. . . . . 6 |- (v = y -> (wAv <-> wAy))
8	5, 6, 7	cbvmo 1412	. . . . 5 |- (E*v wAv <-> E*y wAy)
9	8	albii 1003	. . . 4 |- (A.wE*v wAv <-> A.wE*y wAy)
10		ax-17 975	. . . . . 6 |- (wAv -> A.u wAv)
11	10	mo2 1404	. . . . 5 |- (E*v wAv <-> E.uA.v(wAv -> v = u))
12	11	albii 1003	. . . 4 |- (A.wE*v wAv <-> A.wE.uA.v(wAv -> v = u))
13		ax-17 975	. . . . . . 7 |- (z e. w -> A.x z e. w)
14		dffunmof.1	. . . . . . 7 |- (z e. A -> A.x z e. A)
15		ax-17 975	. . . . . . 7 |- (z e. y -> A.x z e. y)
16	13, 14, 15	hbbr 2673	. . . . . 6 |- (wAy -> A.x wAy)
17	16	hbmo 1411	. . . . 5 |- (E*y wAy -> A.xE*y wAy)
18		ax-17 975	. . . . 5 |- (E*y xAy -> A.wE*y xAy)
19		ax-17 975	. . . . . 6 |- (w = x -> A.y w = x)
20		breq1 2637	. . . . . 6 |- (w = x -> (wAy <-> xAy))
21	19, 20	mobid 1408	. . . . 5 |- (w = x -> (E*y wAy <-> E*y xAy))
22	17, 18, 21	cbval 1169	. . . 4 |- (A.wE*y wAy <-> A.xE*y xAy)
23	9, 12, 22	3bitr3ri 182	. . 3 |- (A.xE*y xAy <-> A.wE.uA.v(wAv -> v = u))
24	23	anbi2i 483	. 2 |- ((Rel A /\ A.xE*y xAy) <-> (Rel A /\ A.wE.uA.v(wAv -> v = u)))
25	1, 24	bitr4i 176	1 |- (Fun A <-> (Rel A /\ A.xE*y xAy))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 958   = wceq 960   e. wcel 962  E.wex 984  E*wmo 1385   class class class wbr 2634  Rel wrel 3191  Fun wfun 3192

This theorem is referenced by:  dffunmo 3547  funopab 3564

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-sep 2718  ax-pow 2758  ax-pr 2795

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-v 1819  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-op 2428  df-br 2635  df-opab 2682  df-id 2851  df-cnv 3202  df-co 3203  df-fun 3208

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