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 ∩ (A × B)) Fn B ↔ ∀y ∈ B ∃!x ∈ 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 ∩ (A × B)) Fn B ↔ (Fun ◡(R ∩ (A × B)) ∧ dom ◡(R ∩ (A × B)) = B))
2		dfrn4 4905	. . . 4 ⊢ ran (R ∩ (A × B)) = dom ◡(R ∩ (A × B))
3	2	eqeq1i 2360	. . 3 ⊢ (ran (R ∩ (A × B)) = B ↔ dom ◡(R ∩ (A × B)) = B)
4	3	anbi2i 675	. 2 ⊢ ((Fun ◡(R ∩ (A × B)) ∧ ran (R ∩ (A × B)) = B) ↔ (Fun ◡(R ∩ (A × B)) ∧ dom ◡(R ∩ (A × B)) = B))
5		rninxp 5061	. . . . 5 ⊢ (ran (R ∩ (A × B)) = B ↔ ∀y ∈ B ∃x ∈ A xRy)
6	5	anbi1i 676	. . . 4 ⊢ ((ran (R ∩ (A × B)) = B ∧ ∀y ∈ B ∃*x ∈ A xRy) ↔ (∀y ∈ B ∃x ∈ A xRy ∧ ∀y ∈ B ∃*x ∈ A xRy))
7		funcnv 5157	. . . . . 6 ⊢ (Fun ◡(R ∩ (A × B)) ↔ ∀y ∈ ran (R ∩ (A × B))∃*x x(R ∩ (A × B))y)
8		raleq 2808	. . . . . . 7 ⊢ (ran (R ∩ (A × B)) = B → (∀y ∈ ran (R ∩ (A × B))∃*x x(R ∩ (A × B))y ↔ ∀y ∈ B ∃*x x(R ∩ (A × B))y))
9		biimt 325	. . . . . . . . 9 ⊢ (y ∈ B → (∃*x ∈ A xRy ↔ (y ∈ B → ∃*x ∈ A xRy)))
10		moanimv 2262	. . . . . . . . . 10 ⊢ (∃*x(y ∈ B ∧ (x ∈ A ∧ xRy)) ↔ (y ∈ B → ∃*x(x ∈ A ∧ xRy)))
11		brin 4694	. . . . . . . . . . . 12 ⊢ (x(R ∩ (A × B))y ↔ (xRy ∧ x(A × B)y))
12		brxp 4813	. . . . . . . . . . . . . . . 16 ⊢ (x(A × B)y ↔ (x ∈ A ∧ y ∈ B))
13		ancom 437	. . . . . . . . . . . . . . . 16 ⊢ ((x ∈ A ∧ y ∈ B) ↔ (y ∈ B ∧ x ∈ A))
14	12, 13	bitri 240	. . . . . . . . . . . . . . 15 ⊢ (x(A × B)y ↔ (y ∈ B ∧ x ∈ A))
15	14	anbi2i 675	. . . . . . . . . . . . . 14 ⊢ ((xRy ∧ x(A × B)y) ↔ (xRy ∧ (y ∈ B ∧ x ∈ A)))
16		ancom 437	. . . . . . . . . . . . . 14 ⊢ ((xRy ∧ (y ∈ B ∧ x ∈ A)) ↔ ((y ∈ B ∧ x ∈ A) ∧ xRy))
17	15, 16	bitri 240	. . . . . . . . . . . . 13 ⊢ ((xRy ∧ x(A × B)y) ↔ ((y ∈ B ∧ x ∈ A) ∧ xRy))
18		anass 630	. . . . . . . . . . . . 13 ⊢ (((y ∈ B ∧ x ∈ A) ∧ xRy) ↔ (y ∈ B ∧ (x ∈ A ∧ xRy)))
19	17, 18	bitri 240	. . . . . . . . . . . 12 ⊢ ((xRy ∧ x(A × B)y) ↔ (y ∈ B ∧ (x ∈ A ∧ xRy)))
20	11, 19	bitri 240	. . . . . . . . . . 11 ⊢ (x(R ∩ (A × B))y ↔ (y ∈ B ∧ (x ∈ A ∧ xRy)))
21	20	mobii 2240	. . . . . . . . . 10 ⊢ (∃*x x(R ∩ (A × B))y ↔ ∃*x(y ∈ B ∧ (x ∈ A ∧ xRy)))
22		df-rmo 2623	. . . . . . . . . . 11 ⊢ (∃*x ∈ A xRy ↔ ∃*x(x ∈ A ∧ xRy))
23	22	imbi2i 303	. . . . . . . . . 10 ⊢ ((y ∈ B → ∃*x ∈ A xRy) ↔ (y ∈ B → ∃*x(x ∈ A ∧ xRy)))
24	10, 21, 23	3bitr4i 268	. . . . . . . . 9 ⊢ (∃*x x(R ∩ (A × B))y ↔ (y ∈ B → ∃*x ∈ A xRy))
25	9, 24	syl6rbbr 255	. . . . . . . 8 ⊢ (y ∈ B → (∃*x x(R ∩ (A × B))y ↔ ∃*x ∈ A xRy))
26	25	ralbiia 2647	. . . . . . 7 ⊢ (∀y ∈ B ∃*x x(R ∩ (A × B))y ↔ ∀y ∈ B ∃*x ∈ A xRy)
27	8, 26	syl6bb 252	. . . . . 6 ⊢ (ran (R ∩ (A × B)) = B → (∀y ∈ ran (R ∩ (A × B))∃*x x(R ∩ (A × B))y ↔ ∀y ∈ B ∃*x ∈ A xRy))
28	7, 27	syl5bb 248	. . . . 5 ⊢ (ran (R ∩ (A × B)) = B → (Fun ◡(R ∩ (A × B)) ↔ ∀y ∈ B ∃*x ∈ A xRy))
29	28	pm5.32i 618	. . . 4 ⊢ ((ran (R ∩ (A × B)) = B ∧ Fun ◡(R ∩ (A × B))) ↔ (ran (R ∩ (A × B)) = B ∧ ∀y ∈ B ∃*x ∈ A xRy))
30		r19.26 2747	. . . 4 ⊢ (∀y ∈ B (∃x ∈ A xRy ∧ ∃*x ∈ A xRy) ↔ (∀y ∈ B ∃x ∈ A xRy ∧ ∀y ∈ B ∃*x ∈ A xRy))
31	6, 29, 30	3bitr4i 268	. . 3 ⊢ ((ran (R ∩ (A × B)) = B ∧ Fun ◡(R ∩ (A × B))) ↔ ∀y ∈ B (∃x ∈ A xRy ∧ ∃*x ∈ A xRy))
32		ancom 437	. . 3 ⊢ ((Fun ◡(R ∩ (A × B)) ∧ ran (R ∩ (A × B)) = B) ↔ (ran (R ∩ (A × B)) = B ∧ Fun ◡(R ∩ (A × B))))
33		reu5 2825	. . . 4 ⊢ (∃!x ∈ A xRy ↔ (∃x ∈ A xRy ∧ ∃*x ∈ A xRy))
34	33	ralbii 2639	. . 3 ⊢ (∀y ∈ B ∃!x ∈ A xRy ↔ ∀y ∈ B (∃x ∈ A xRy ∧ ∃*x ∈ A xRy))
35	31, 32, 34	3bitr4i 268	. 2 ⊢ ((Fun ◡(R ∩ (A × B)) ∧ ran (R ∩ (A × B)) = B) ↔ ∀y ∈ B ∃!x ∈ A xRy)
36	1, 4, 35	3bitr2i 264	1 ⊢ (◡(R ∩ (A × B)) Fn B ↔ ∀y ∈ B ∃!x ∈ A xRy)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2615  ∃wrex 2616  ∃!wreu 2617  ∃*wrmo 2618   ∩ cin 3209   class class class wbr 4640   × 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)