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Theorem fncnv 5158
 Description: Single-rootedness (see funcnv 5156) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)
Assertion
Ref Expression
fncnv ((R ∩ (A × B)) Fn By B ∃!x A xRy)
Distinct variable groups:   x,y,A   x,B,y   x,R,y

Proof of Theorem fncnv
StepHypRef Expression
1 df-fn 4790 . 2 ((R ∩ (A × B)) Fn B ↔ (Fun (R ∩ (A × B)) dom (R ∩ (A × B)) = B))
2 dfrn4 4904 . . . 4 ran (R ∩ (A × B)) = dom (R ∩ (A × B))
32eqeq1i 2360 . . 3 (ran (R ∩ (A × B)) = B ↔ dom (R ∩ (A × B)) = B)
43anbi2i 675 . 2 ((Fun (R ∩ (A × B)) ran (R ∩ (A × B)) = B) ↔ (Fun (R ∩ (A × B)) dom (R ∩ (A × B)) = B))
5 rninxp 5060 . . . . 5 (ran (R ∩ (A × B)) = By B x A xRy)
65anbi1i 676 . . . 4 ((ran (R ∩ (A × B)) = B y B ∃*x A xRy) ↔ (y B x A xRy y B ∃*x A xRy))
7 funcnv 5156 . . . . . 6 (Fun (R ∩ (A × B)) ↔ y ran (R ∩ (A × B))∃*x x(R ∩ (A × B))y)
8 raleq 2807 . . . . . . 7 (ran (R ∩ (A × B)) = B → (y ran (R ∩ (A × B))∃*x x(R ∩ (A × B))yy 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 4693 . . . . . . . . . . . 12 (x(R ∩ (A × B))y ↔ (xRy x(A × B)y))
12 brxp 4812 . . . . . . . . . . . . . . . 16 (x(A × B)y ↔ (x A y B))
13 ancom 437 . . . . . . . . . . . . . . . 16 ((x A y B) ↔ (y B x A))
1412, 13bitri 240 . . . . . . . . . . . . . . 15 (x(A × B)y ↔ (y B x A))
1514anbi2i 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))
1715, 16bitri 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)))
1917, 18bitri 240 . . . . . . . . . . . 12 ((xRy x(A × B)y) ↔ (y B (x A xRy)))
2011, 19bitri 240 . . . . . . . . . . 11 (x(R ∩ (A × B))y ↔ (y B (x A xRy)))
2120mobii 2240 . . . . . . . . . 10 (∃*x x(R ∩ (A × B))y∃*x(y B (x A xRy)))
22 df-rmo 2622 . . . . . . . . . . 11 (∃*x A xRy∃*x(x A xRy))
2322imbi2i 303 . . . . . . . . . 10 ((y B∃*x A xRy) ↔ (y B∃*x(x A xRy)))
2410, 21, 233bitr4i 268 . . . . . . . . 9 (∃*x x(R ∩ (A × B))y ↔ (y B∃*x A xRy))
259, 24syl6rbbr 255 . . . . . . . 8 (y B → (∃*x x(R ∩ (A × B))y∃*x A xRy))
2625ralbiia 2646 . . . . . . 7 (y B ∃*x x(R ∩ (A × B))yy B ∃*x A xRy)
278, 26syl6bb 252 . . . . . 6 (ran (R ∩ (A × B)) = B → (y ran (R ∩ (A × B))∃*x x(R ∩ (A × B))yy B ∃*x A xRy))
287, 27syl5bb 248 . . . . 5 (ran (R ∩ (A × B)) = B → (Fun (R ∩ (A × B)) ↔ y B ∃*x A xRy))
2928pm5.32i 618 . . . 4 ((ran (R ∩ (A × B)) = B Fun (R ∩ (A × B))) ↔ (ran (R ∩ (A × B)) = B y B ∃*x A xRy))
30 r19.26 2746 . . . 4 (y B (x A xRy ∃*x A xRy) ↔ (y B x A xRy y B ∃*x A xRy))
316, 29, 303bitr4i 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 2824 . . . 4 (∃!x A xRy ↔ (x A xRy ∃*x A xRy))
3433ralbii 2638 . . 3 (y B ∃!x A xRyy B (x A xRy ∃*x A xRy))
3531, 32, 343bitr4i 268 . 2 ((Fun (R ∩ (A × B)) ran (R ∩ (A × B)) = B) ↔ y B ∃!x A xRy)
361, 4, 353bitr2i 264 1 ((R ∩ (A × B)) Fn By B ∃!x A xRy)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616  ∃*wrmo 2617   ∩ cin 3208   class class class wbr 4639   × cxp 4770  ◡ccnv 4771  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn wfn 4776 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790 This theorem is referenced by: (None)
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