Theorem fncnv 4479

Description: Single-rootedness (see funcnv 4475) 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 4009	. 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 4005	. . . 4 |- ran ( R i^i (A X. B)) = dom `'(R i^i (A X. B))
3	2	eqeq1i 1891	. . 3 |- (ran ( R i^i (A X. B)) = B <-> dom `'(R i^i (A X. B)) = B)
4	3	anbi2i 538	. 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 4355	. . . . 5 |- (ran ( R i^i (A X. B)) = B <-> A.y e. B E.x e. A xRy)
6	5	anbi1i 539	. . . 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		raleq 2266	. . . . . . 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 803	. . . . . . . . 9 |- (y e. B -> (E*x(x e. A /\ xRy) <-> (y e. B -> E*x(x e. A /\ xRy))))
9		visset 2295	. . . . . . . . . . . . 13 |- y e. _V
10		brinxp2 4057	. . . . . . . . . . . . 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 865	. . . . . . . . . . . 12 |- ((x e. A /\ y e. B /\ xRy) <-> (y e. B /\ x e. A /\ xRy))
13		3anass 862	. . . . . . . . . . . 12 |- ((y e. B /\ x e. A /\ xRy) <-> (y e. B /\ (x e. A /\ xRy)))
14	11, 12, 13	3bitri 194	. . . . . . . . . . 11 |- (x(R i^i (A X. B))y <-> (y e. B /\ (x e. A /\ xRy)))
15	14	mobii 1801	. . . . . . . . . 10 |- (E*x x(R i^i (A X. B))y <-> E*x(y e. B /\ (x e. A /\ xRy)))
16		moanimv 1829	. . . . . . . . . 10 |- (E*x(y e. B /\ (x e. A /\ xRy)) <-> (y e. B -> E*x(x e. A /\ xRy)))
17	15, 16	bitri 190	. . . . . . . . 9 |- (E*x x(R i^i (A X. B))y <-> (y e. B -> E*x(x e. A /\ xRy)))
18	8, 17	syl6rbbr 598	. . . . . . . 8 |- (y e. B -> (E*x x(R i^i (A X. B))y <-> E*x(x e. A /\ xRy)))
19	18	ralbiia 2133	. . . . . . 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 595	. . . . . 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 4475	. . . . . 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 591	. . . . 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 707	. . . 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 2219	. . . 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	3bitr4i 200	. . 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 482	. . 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 2441	. . . 4 |- (E!x e. A xRy <-> (E.x e. A xRy /\ E*x(x e. A /\ xRy)))
28	27	ralbii 2127	. . 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	3bitr4i 200	. 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	3bitr2i 196	1 |- (`'(R i^i (A X. B)) Fn B <-> A.y e. B E!x e. A xRy)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E*wmo 1772  A.wral 2105  E.wrex 2106  E!wreu 2107  _Vcvv 2292   i^i cin 2592   class class class wbr 3338   X. cxp 3984  `'ccnv 3985  dom cdm 3986  ran crn 3987  Fun wfun 3992   Fn wfn 3993

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009

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