Theorem dffunmof 3522

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 3519	. 2 |- (Fun A <-> (Rel A /\ A.wE.uA.v(wAv -> v = u)))
2		ax-17 969	. . . . . . 7 |- (z e. w -> A.y z e. w)
3		dffunmof.2	. . . . . . 7 |- (z e. A -> A.y z e. A)
4		ax-17 969	. . . . . . 7 |- (z e. v -> A.y z e. v)
5	2, 3, 4	hbbr 2653	. . . . . 6 |- (wAv -> A.y wAv)
6		ax-17 969	. . . . . 6 |- (wAy -> A.v wAy)
7		breq2 2618	. . . . . 6 |- (v = y -> (wAv <-> wAy))
8	5, 6, 7	cbvmo 1406	. . . . 5 |- (E*v wAv <-> E*y wAy)
9	8	albii 997	. . . 4 |- (A.wE*v wAv <-> A.wE*y wAy)
10		ax-17 969	. . . . . 6 |- (wAv -> A.u wAv)
11	10	mo2 1398	. . . . 5 |- (E*v wAv <-> E.uA.v(wAv -> v = u))
12	11	albii 997	. . . 4 |- (A.wE*v wAv <-> A.wE.uA.v(wAv -> v = u))
13		ax-17 969	. . . . . . 7 |- (z e. w -> A.x z e. w)
14		dffunmof.1	. . . . . . 7 |- (z e. A -> A.x z e. A)
15		ax-17 969	. . . . . . 7 |- (z e. y -> A.x z e. y)
16	13, 14, 15	hbbr 2653	. . . . . 6 |- (wAy -> A.x wAy)
17	16	hbmo 1405	. . . . 5 |- (E*y wAy -> A.xE*y wAy)
18		ax-17 969	. . . . 5 |- (E*y xAy -> A.wE*y xAy)
19		ax-17 969	. . . . . 6 |- (w = x -> A.y w = x)
20		breq1 2617	. . . . . 6 |- (w = x -> (wAy <-> xAy))
21	19, 20	mobid 1402	. . . . 5 |- (w = x -> (E*y wAy <-> E*y xAy))
22	17, 18, 21	cbval 1163	. . . 4 |- (A.wE*y wAy <-> A.xE*y xAy)
23	9, 12, 22	3bitr3r 182	. . 3 |- (A.xE*y xAy <-> A.wE.uA.v(wAv -> v = u))
24	23	anbi2i 480	. 2 |- ((Rel A /\ A.xE*y xAy) <-> (Rel A /\ A.wE.uA.v(wAv -> v = u)))
25	1, 24	bitr4 176	1 |- (Fun A <-> (Rel A /\ A.xE*y xAy))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  E*wmo 1379   class class class wbr 2614  Rel wrel 3170  Fun wfun 3171

This theorem is referenced by:  dffunmo 3523  funopab 3540

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-cnv 3181  df-co 3182  df-fun 3187

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