Theorem fncnv 3636

Description: Single-rootedness (see funcnv 3632) 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 3248	. 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 3244	. . . 4 |- ran ( R i^i (A X. B)) = dom `'(R i^i (A X. B))
3	2	eqeq1i 1519	. . 3 |- (ran ( R i^i (A X. B)) = B <-> dom `'(R i^i (A X. B)) = B)
4	3	anbi2i 482	. 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 3538	. . . . 5 |- (ran ( R i^i (A X. B)) = B <-> A.y e. B E.x e. A xRy)
6	5	anbi1i 483	. . . 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 1824	. . . . . . 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 734	. . . . . . . . 9 |- (y e. B -> (E*x(x e. A /\ xRy) <-> (y e. B -> E*x(x e. A /\ xRy))))
9		visset 1851	. . . . . . . . . . . . 13 |- y e. V
10		brinxp2 3290	. . . . . . . . . . . . 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 785	. . . . . . . . . . . 12 |- ((x e. A /\ y e. B /\ xRy) <-> (y e. B /\ x e. A /\ xRy))
13		3anass 782	. . . . . . . . . . . 12 |- ((y e. B /\ x e. A /\ xRy) <-> (y e. B /\ (x e. A /\ xRy)))
14	11, 12, 13	3bitri 175	. . . . . . . . . . 11 |- (x(R i^i (A X. B))y <-> (y e. B /\ (x e. A /\ xRy)))
15	14	mobii 1438	. . . . . . . . . 10 |- (E*x x(R i^i (A X. B))y <-> E*x(y e. B /\ (x e. A /\ xRy)))
16		moanimv 1462	. . . . . . . . . 10 |- (E*x(y e. B /\ (x e. A /\ xRy)) <-> (y e. B -> E*x(x e. A /\ xRy)))
17	15, 16	bitri 171	. . . . . . . . 9 |- (E*x x(R i^i (A X. B))y <-> (y e. B -> E*x(x e. A /\ xRy)))
18	8, 17	syl6rbbr 541	. . . . . . . 8 |- (y e. B -> (E*x x(R i^i (A X. B))y <-> E*x(x e. A /\ xRy)))
19	18	ralbiia 1711	. . . . . . 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 538	. . . . . 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 3632	. . . . . 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 534	. . . . 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 647	. . . 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 1788	. . . 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 181	. . 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 437	. . 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 1967	. . . 4 |- (E!x e. A xRy <-> (E.x e. A xRy /\ E*x(x e. A /\ xRy)))
28	27	ralbii 1705	. . 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 181	. 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 177	1 |- (`'(R i^i (A X. B)) Fn B <-> A.y e. B E!x e. A xRy)

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   /\ w3a 778   = wceq 988   e. wcel 990  E*wmo 1414  A.wral 1683  E.wrex 1684  E!wreu 1685  Vcvv 1849   i^i cin 2090   class class class wbr 2669   X. cxp 3223  `'ccnv 3224  dom cdm 3225  ran crn 3226  Fun wfun 3231   Fn wfn 3232

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-reu 1689  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-fun 3247  df-fn 3248

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