Theorem fncnv 5359

Description: Single-rootedness (see funcnv 5354) 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 5293	. 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 4704	. . . 4 |- ran ( R i^i ( A X. B ) ) = dom `' ( R i^i ( A X. B ) )
3	2	eqeq1i 2215	. . 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 5145	. . . . 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 5354	. . . . . 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 2705	. . . . . . 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 2131	. . . . . . . . . 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 4760	. . . . . . . . . . . 12 |- ( x ( R i^i ( A X. B ) ) y <-> ( x e. A /\ y e. B /\ x R y ) )
11		3anan12 993	. . . . . . . . . . . 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 2092	. . . . . . . . . 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 2494	. . . . . . . . . . 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 2522	. . . . . . 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 2634	. . . 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 2726	. . . 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 2514	. . 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 981   = wceq 1373   E*wmo 2056   e. wcel 2178   A.wral 2486   E.wrex 2487   E!wreu 2488   E*wrmo 2489   i^i cin 3173   class class class wbr 4059   X. cxp 4691   `'ccnv 4692   dom cdm 4693   ran crn 4694   Fun wfun 5284   Fn wfn 5285

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-fun 5292  df-fn 5293

This theorem is referenced by: (None)