Theorem fncnv 5403

Description: Single-rootedness (see funcnv 5398) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)

Assertion

Ref	Expression
fncnv	|- ( `' ( R i^i ( A X. B ) ) Fn B <-> A. y e. B E! x e. A x R y )

Distinct variable groups:   x, y, A   x, B, y   x, R, y

Proof of Theorem fncnv

Step	Hyp	Ref	Expression
1		df-fn 5336	. 2 |- ( `' ( R i^i ( A X. B ) ) Fn B <-> ( Fun `' ( R i^i ( A X. B ) ) /\ dom `' ( R i^i ( A X. B ) ) = B ) )
2		df-rn 4742	. . . 4 |- ran ( R i^i ( A X. B ) ) = dom `' ( R i^i ( A X. B ) )
3	2	eqeq1i 2239	. . 3 |- ( ran ( R i^i ( A X. B ) ) = B <-> dom `' ( R i^i ( A X. B ) ) = B )
4	3	anbi2i 457	. 2 |- ( ( Fun `' ( R i^i ( A X. B ) ) /\ ran ( R i^i ( A X. B ) ) = B ) <-> ( Fun `' ( R i^i ( A X. B ) ) /\ dom `' ( R i^i ( A X. B ) ) = B ) )
5		rninxp 5187	. . . . 5 |- ( ran ( R i^i ( A X. B ) ) = B <-> A. y e. B E. x e. A x R y )
6	5	anbi1i 458	. . . 4 |- ( ( ran ( R i^i ( A X. B ) ) = B /\ A. y e. B E* x e. A x R y ) <-> ( A. y e. B E. x e. A x R y /\ A. y e. B E* x e. A x R y ) )
7		funcnv 5398	. . . . . 6 |- ( Fun `' ( R i^i ( A X. B ) ) <-> A. y e. ran ( R i^i ( A X. B ) ) E* x x ( R i^i ( A X. B ) ) y )
8		raleq 2731	. . . . . . 7 |- ( ran ( R i^i ( A X. B ) ) = B -> ( A. y e. ran ( R i^i ( A X. B ) ) E* x x ( R i^i ( A X. B ) ) y <-> A. y e. B E* x x ( R i^i ( A X. B ) ) y ) )
9		moanimv 2155	. . . . . . . . . 10 |- ( E* x ( y e. B /\ ( x e. A /\ x R y ) ) <-> ( y e. B -> E* x ( x e. A /\ x R y ) ) )
10		brinxp2 4799	. . . . . . . . . . . 12 |- ( x ( R i^i ( A X. B ) ) y <-> ( x e. A /\ y e. B /\ x R y ) )
11		3anan12 1017	. . . . . . . . . . . 12 |- ( ( x e. A /\ y e. B /\ x R y ) <-> ( y e. B /\ ( x e. A /\ x R y ) ) )
12	10, 11	bitri 184	. . . . . . . . . . 11 |- ( x ( R i^i ( A X. B ) ) y <-> ( y e. B /\ ( x e. A /\ x R y ) ) )
13	12	mobii 2116	. . . . . . . . . 10 |- ( E* x x ( R i^i ( A X. B ) ) y <-> E* x ( y e. B /\ ( x e. A /\ x R y ) ) )
14		df-rmo 2519	. . . . . . . . . . 11 |- ( E* x e. A x R y <-> E* x ( x e. A /\ x R y ) )
15	14	imbi2i 226	. . . . . . . . . 10 |- ( ( y e. B -> E* x e. A x R y ) <-> ( y e. B -> E* x ( x e. A /\ x R y ) ) )
16	9, 13, 15	3bitr4i 212	. . . . . . . . 9 |- ( E* x x ( R i^i ( A X. B ) ) y <-> ( y e. B -> E* x e. A x R y ) )
17		biimt 241	. . . . . . . . 9 |- ( y e. B -> ( E* x e. A x R y <-> ( y e. B -> E* x e. A x R y ) ) )
18	16, 17	bitr4id 199	. . . . . . . 8 |- ( y e. B -> ( E* x x ( R i^i ( A X. B ) ) y <-> E* x e. A x R y ) )
19	18	ralbiia 2547	. . . . . . 7 |- ( A. y e. B E* x x ( R i^i ( A X. B ) ) y <-> A. y e. B E* x e. A x R y )
20	8, 19	bitrdi 196	. . . . . 6 |- ( ran ( R i^i ( A X. B ) ) = B -> ( A. y e. ran ( R i^i ( A X. B ) ) E* x x ( R i^i ( A X. B ) ) y <-> A. y e. B E* x e. A x R y ) )
21	7, 20	bitrid 192	. . . . 5 |- ( ran ( R i^i ( A X. B ) ) = B -> ( Fun `' ( R i^i ( A X. B ) ) <-> A. y e. B E* x e. A x R y ) )
22	21	pm5.32i 454	. . . 4 |- ( ( ran ( R i^i ( A X. B ) ) = B /\ Fun `' ( R i^i ( A X. B ) ) ) <-> ( ran ( R i^i ( A X. B ) ) = B /\ A. y e. B E* x e. A x R y ) )
23		r19.26 2660	. . . 4 |- ( A. y e. B ( E. x e. A x R y /\ E* x e. A x R y ) <-> ( A. y e. B E. x e. A x R y /\ A. y e. B E* x e. A x R y ) )
24	6, 22, 23	3bitr4i 212	. . 3 |- ( ( ran ( R i^i ( A X. B ) ) = B /\ Fun `' ( R i^i ( A X. B ) ) ) <-> A. y e. B ( E. x e. A x R y /\ E* x e. A x R y ) )
25		ancom 266	. . 3 |- ( ( Fun `' ( R i^i ( A X. B ) ) /\ ran ( R i^i ( A X. B ) ) = B ) <-> ( ran ( R i^i ( A X. B ) ) = B /\ Fun `' ( R i^i ( A X. B ) ) ) )
26		reu5 2752	. . . 4 |- ( E! x e. A x R y <-> ( E. x e. A x R y /\ E* x e. A x R y ) )
27	26	ralbii 2539	. . 3 |- ( A. y e. B E! x e. A x R y <-> A. y e. B ( E. x e. A x R y /\ E* x e. A x R y ) )
28	24, 25, 27	3bitr4i 212	. 2 |- ( ( Fun `' ( R i^i ( A X. B ) ) /\ ran ( R i^i ( A X. B ) ) = B ) <-> A. y e. B E! x e. A x R y )
29	1, 4, 28	3bitr2i 208	1 |- ( `' ( R i^i ( A X. B ) ) Fn B <-> A. y e. B E! x e. A x R y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   E*wmo 2080   e. wcel 2202   A.wral 2511   E.wrex 2512   E!wreu 2513   E*wrmo 2514   i^i cin 3200   class class class wbr 4093   X. cxp 4729   `'ccnv 4730   dom cdm 4731   ran crn 4732   Fun wfun 5327   Fn wfn 5328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-fun 5335  df-fn 5336

This theorem is referenced by: (None)