Theorem fncnv 3547

Description: Single-rootedness (see funcnv 3543) of a class cut down by a cross product.

Assertion

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

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

Proof of Theorem fncnv

Step	Hyp	Ref	Expression
1		df-fn 3183	. 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 3179	. . . 4 |- ran ( R i^i (A X. B)) = dom `'(R i^i (A X. B))
3	2	eqeq1i 1474	. . 3 |- (ran ( R i^i (A X. B)) = B <-> dom `'(R i^i (A X. B)) = B)
4	3	anbi2i 479	. 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 3468	. . . . 5 |- (ran ( R i^i (A X. B)) = B <-> A.y e. B E.x e. A xRy)
6	5	anbi1i 480	. . . 4 |- ((ran ( R i^i (A X. B)) = B /\ A.y e. B E*x(x e. A /\ xRy)) <-> (A.y e. B E.x e. A xRy /\ A.y e. B E*x(x e. A /\ xRy)))
7		raleq1 1778	. . . . . . 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))
8		biimt 729	. . . . . . . . 9 |- (y e. B -> (E*x(x e. A /\ xRy) <-> (y e. B -> E*x(x e. A /\ xRy))))
9		visset 1804	. . . . . . . . . . . . 13 |- y e. V
10		brinxp2 3221	. . . . . . . . . . . . 13 |- (y e. V -> (x(R i^i (A X. B))y <-> (x e. A /\ y e. B /\ xRy)))
11	9, 10	ax-mp 7	. . . . . . . . . . . 12 |- (x(R i^i (A X. B))y <-> (x e. A /\ y e. B /\ xRy))
12		3ancoma 780	. . . . . . . . . . . 12 |- ((x e. A /\ y e. B /\ xRy) <-> (y e. B /\ x e. A /\ xRy))
13		3anass 777	. . . . . . . . . . . 12 |- ((y e. B /\ x e. A /\ xRy) <-> (y e. B /\ (x e. A /\ xRy)))
14	11, 12, 13	3bitr 177	. . . . . . . . . . 11 |- (x(R i^i (A X. B))y <-> (y e. B /\ (x e. A /\ xRy)))
15	14	mobii 1398	. . . . . . . . . 10 |- (E*x x(R i^i (A X. B))y <-> E*x(y e. B /\ (x e. A /\ xRy)))
16		moanimv 1422	. . . . . . . . . 10 |- (E*x(y e. B /\ (x e. A /\ xRy)) <-> (y e. B -> E*x(x e. A /\ xRy)))
17	15, 16	bitr 173	. . . . . . . . 9 |- (E*x x(R i^i (A X. B))y <-> (y e. B -> E*x(x e. A /\ xRy)))
18	8, 17	syl6rbbr 537	. . . . . . . 8 |- (y e. B -> (E*x x(R i^i (A X. B))y <-> E*x(x e. A /\ xRy)))
19	18	ralbiia 1665	. . . . . . 7 |- (A.y e. B E*x x(R i^i (A X. B))y <-> A.y e. B E*x(x e. A /\ xRy))
20	7, 19	syl6bb 534	. . . . . 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(x e. A /\ xRy)))
21		funcnv 3543	. . . . . 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)
22	20, 21	syl5bb 530	. . . . 5 |- (ran ( R i^i (A X. B)) = B -> (Fun `'(R i^i (A X. B)) <-> A.y e. B E*x(x e. A /\ xRy)))
23	22	pm5.32i 643	. . . 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(x e. A /\ xRy)))
24		r19.26 1742	. . . 4 |- (A.y e. B (E.x e. A xRy /\ E*x(x e. A /\ xRy)) <-> (A.y e. B E.x e. A xRy /\ A.y e. B E*x(x e. A /\ xRy)))
25	6, 23, 24	3bitr4 183	. . 3 |- ((ran ( R i^i (A X. B)) = B /\ Fun `'(R i^i (A X. B))) <-> A.y e. B (E.x e. A xRy /\ E*x(x e. A /\ xRy)))
26		ancom 435	. . 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))))
27		reu5 1919	. . . 4 |- (E!x e. A xRy <-> (E.x e. A xRy /\ E*x(x e. A /\ xRy)))
28	27	ralbii 1659	. . 3 |- (A.y e. B E!x e. A xRy <-> A.y e. B (E.x e. A xRy /\ E*x(x e. A /\ xRy)))
29	25, 26, 28	3bitr4 183	. 2 |- ((Fun `'(R i^i (A X. B)) /\ ran ( R i^i (A X. B)) = B) <-> A.y e. B E!x e. A xRy)
30	1, 4, 29	3bitr2 179	1 |- (`'(R i^i (A X. B)) Fn B <-> A.y e. B E!x e. A xRy)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  E*wmo 1374  A.wral 1637  E.wrex 1638  E!wreu 1639  Vcvv 1802   i^i cin 2036   class class class wbr 2609   X. cxp 3158  `'ccnv 3159  dom cdm 3160  ran crn 3161  Fun wfun 3166   Fn wfn 3167

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-fun 3182  df-fn 3183

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