Theorem fncnv 5159

Description: Single-rootedness (see funcnv 5157) 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 xRy)

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

Proof of Theorem fncnv

Step	Hyp	Ref	Expression
1		df-fn 4791	. 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		dfrn4 4905	. . . 4 |- ran (R i^i (A X. B)) = dom `'(R i^i (A X. B))
3	2	eqeq1i 2360	. . 3 |- (ran (R i^i (A X. B)) = B <-> dom `'(R i^i (A X. B)) = B)
4	3	anbi2i 675	. 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 5061	. . . . 5 |- (ran (R i^i (A X. B)) = B <-> A.y e. B E.x e. A xRy)
6	5	anbi1i 676	. . . 4 |- ((ran (R i^i (A X. B)) = B /\ A.y e. B E*x e. A xRy) <-> (A.y e. B E.x e. A xRy /\ A.y e. B E*x e. A xRy))
7		funcnv 5157	. . . . . 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 2808	. . . . . . 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		biimt 325	. . . . . . . . 9 |- (y e. B -> (E*x e. A xRy <-> (y e. B -> E*x e. A xRy)))
10		moanimv 2262	. . . . . . . . . 10 |- (E*x(y e. B /\ (x e. A /\ xRy)) <-> (y e. B -> E*x(x e. A /\ xRy)))
11		brin 4694	. . . . . . . . . . . 12 |- (x(R i^i (A X. B))y <-> (xRy /\ x(A X. B)y))
12		brxp 4813	. . . . . . . . . . . . . . . 16 |- (x(A X. B)y <-> (x e. A /\ y e. B))
13		ancom 437	. . . . . . . . . . . . . . . 16 |- ((x e. A /\ y e. B) <-> (y e. B /\ x e. A))
14	12, 13	bitri 240	. . . . . . . . . . . . . . 15 |- (x(A X. B)y <-> (y e. B /\ x e. A))
15	14	anbi2i 675	. . . . . . . . . . . . . 14 |- ((xRy /\ x(A X. B)y) <-> (xRy /\ (y e. B /\ x e. A)))
16		ancom 437	. . . . . . . . . . . . . 14 |- ((xRy /\ (y e. B /\ x e. A)) <-> ((y e. B /\ x e. A) /\ xRy))
17	15, 16	bitri 240	. . . . . . . . . . . . 13 |- ((xRy /\ x(A X. B)y) <-> ((y e. B /\ x e. A) /\ xRy))
18		anass 630	. . . . . . . . . . . . 13 |- (((y e. B /\ x e. A) /\ xRy) <-> (y e. B /\ (x e. A /\ xRy)))
19	17, 18	bitri 240	. . . . . . . . . . . 12 |- ((xRy /\ x(A X. B)y) <-> (y e. B /\ (x e. A /\ xRy)))
20	11, 19	bitri 240	. . . . . . . . . . 11 |- (x(R i^i (A X. B))y <-> (y e. B /\ (x e. A /\ xRy)))
21	20	mobii 2240	. . . . . . . . . 10 |- (E*x x(R i^i (A X. B))y <-> E*x(y e. B /\ (x e. A /\ xRy)))
22		df-rmo 2623	. . . . . . . . . . 11 |- (E*x e. A xRy <-> E*x(x e. A /\ xRy))
23	22	imbi2i 303	. . . . . . . . . 10 |- ((y e. B -> E*x e. A xRy) <-> (y e. B -> E*x(x e. A /\ xRy)))
24	10, 21, 23	3bitr4i 268	. . . . . . . . 9 |- (E*x x(R i^i (A X. B))y <-> (y e. B -> E*x e. A xRy))
25	9, 24	syl6rbbr 255	. . . . . . . 8 |- (y e. B -> (E*x x(R i^i (A X. B))y <-> E*x e. A xRy))
26	25	ralbiia 2647	. . . . . . 7 |- (A.y e. B E*x x(R i^i (A X. B))y <-> A.y e. B E*x e. A xRy)
27	8, 26	syl6bb 252	. . . . . 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 xRy))
28	7, 27	syl5bb 248	. . . . 5 |- (ran (R i^i (A X. B)) = B -> (Fun `'(R i^i (A X. B)) <-> A.y e. B E*x e. A xRy))
29	28	pm5.32i 618	. . . 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 xRy))
30		r19.26 2747	. . . 4 |- (A.y e. B (E.x e. A xRy /\ E*x e. A xRy) <-> (A.y e. B E.x e. A xRy /\ A.y e. B E*x e. A xRy))
31	6, 29, 30	3bitr4i 268	. . 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 e. A xRy))
32		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))))
33		reu5 2825	. . . 4 |- (E!x e. A xRy <-> (E.x e. A xRy /\ E*x e. A xRy))
34	33	ralbii 2639	. . 3 |- (A.y e. B E!x e. A xRy <-> A.y e. B (E.x e. A xRy /\ E*x e. A xRy))
35	31, 32, 34	3bitr4i 268	. 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)
36	1, 4, 35	3bitr2i 264	1 |- (`'(R i^i (A X. B)) Fn B <-> A.y e. B E!x e. A xRy)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  E*wmo 2205  A.wral 2615  E.wrex 2616  E!wreu 2617  E*wrmo 2618   i^i cin 3209   class class class wbr 4640   X. cxp 4771  `'ccnv 4772  dom cdm 4773  ran crn 4774  Fun wfun 4776   Fn wfn 4777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791

This theorem is referenced by: (None)