Theorem fnfullfunlem1 5857

Description: Lemma for fnfullfun 5859. Binary relationship over part one of the full function definition. (Contributed by SF, 9-Mar-2015.)

Assertion

Ref	Expression
fnfullfunlem1	|- (A(( _I o. F) \ ( ∼ _I o. F))B <-> (AFB /\ A.x(AFx -> x = B)))

Distinct variable groups:   x,A   x,B   x,F

Proof of Theorem fnfullfunlem1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. . 3 |- (A(( _I o. F) \ ( ∼ _I o. F))B -> (A e. _V /\ B e. _V))
2	1	simprd 449	. 2 |- (A(( _I o. F) \ ( ∼ _I o. F))B -> B e. _V)
3		brex 4690	. . . 4 |- (AFB -> (A e. _V /\ B e. _V))
4	3	simprd 449	. . 3 |- (AFB -> B e. _V)
5	4	adantr 451	. 2 |- ((AFB /\ A.x(AFx -> x = B)) -> B e. _V)
6		breq2 4644	. . . 4 |- (y = B -> (A(( _I o. F) \ ( ∼ _I o. F))y <-> A(( _I o. F) \ ( ∼ _I o. F))B))
7		breq2 4644	. . . . 5 |- (y = B -> (AFy <-> AFB))
8		eqeq2 2362	. . . . . . 7 |- (y = B -> (x = y <-> x = B))
9	8	imbi2d 307	. . . . . 6 |- (y = B -> ((AFx -> x = y) <-> (AFx -> x = B)))
10	9	albidv 1625	. . . . 5 |- (y = B -> (A.x(AFx -> x = y) <-> A.x(AFx -> x = B)))
11	7, 10	anbi12d 691	. . . 4 |- (y = B -> ((AFy /\ A.x(AFx -> x = y)) <-> (AFB /\ A.x(AFx -> x = B))))
12	6, 11	bibi12d 312	. . 3 |- (y = B -> ((A(( _I o. F) \ ( ∼ _I o. F))y <-> (AFy /\ A.x(AFx -> x = y))) <-> (A(( _I o. F) \ ( ∼ _I o. F))B <-> (AFB /\ A.x(AFx -> x = B)))))
13		brdif 4695	. . . 4 |- (A(( _I o. F) \ ( ∼ _I o. F))y <-> (A( _I o. F)y /\ -. A( ∼ _I o. F)y))
14		coi2 5096	. . . . . 6 |- ( _I o. F) = F
15	14	breqi 4646	. . . . 5 |- (A( _I o. F)y <-> AFy)
16		brco 4884	. . . . . . 7 |- (A( ∼ _I o. F)y <-> E.x(AFx /\ x ∼ _I y))
17		df-br 4641	. . . . . . . . . 10 |- (x ∼ _I y <-> <.x, y>. e. ∼ _I )
18		vex 2863	. . . . . . . . . . . . 13 |- x e. _V
19		vex 2863	. . . . . . . . . . . . 13 |- y e. _V
20	18, 19	opex 4589	. . . . . . . . . . . 12 |- <.x, y>. e. _V
21	20	elcompl 3226	. . . . . . . . . . 11 |- (<.x, y>. e. ∼ _I <-> -. <.x, y>. e. _I )
22		df-br 4641	. . . . . . . . . . . 12 |- (x _I y <-> <.x, y>. e. _I )
23	19	ideq 4871	. . . . . . . . . . . 12 |- (x _I y <-> x = y)
24	22, 23	bitr3i 242	. . . . . . . . . . 11 |- (<.x, y>. e. _I <-> x = y)
25	21, 24	xchbinx 301	. . . . . . . . . 10 |- (<.x, y>. e. ∼ _I <-> -. x = y)
26	17, 25	bitri 240	. . . . . . . . 9 |- (x ∼ _I y <-> -. x = y)
27	26	anbi2i 675	. . . . . . . 8 |- ((AFx /\ x ∼ _I y) <-> (AFx /\ -. x = y))
28	27	exbii 1582	. . . . . . 7 |- (E.x(AFx /\ x ∼ _I y) <-> E.x(AFx /\ -. x = y))
29		exanali 1585	. . . . . . 7 |- (E.x(AFx /\ -. x = y) <-> -. A.x(AFx -> x = y))
30	16, 28, 29	3bitrri 263	. . . . . 6 |- (-. A.x(AFx -> x = y) <-> A( ∼ _I o. F)y)
31	30	con1bii 321	. . . . 5 |- (-. A( ∼ _I o. F)y <-> A.x(AFx -> x = y))
32	15, 31	anbi12i 678	. . . 4 |- ((A( _I o. F)y /\ -. A( ∼ _I o. F)y) <-> (AFy /\ A.x(AFx -> x = y)))
33	13, 32	bitri 240	. . 3 |- (A(( _I o. F) \ ( ∼ _I o. F))y <-> (AFy /\ A.x(AFx -> x = y)))
34	12, 33	vtoclg 2915	. 2 |- (B e. _V -> (A(( _I o. F) \ ( ∼ _I o. F))B <-> (AFB /\ A.x(AFx -> x = B))))
35	2, 5, 34	pm5.21nii 342	1 |- (A(( _I o. F) \ ( ∼ _I o. F))B <-> (AFB /\ A.x(AFx -> x = B)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860   ∼ ccompl 3206   \ cdif 3207  <.cop 4562   class class class wbr 4640   o. ccom 4722   _I cid 4764

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-id 4768  df-cnv 4786

This theorem is referenced by:  fnfullfunlem2  5858  fvfullfunlem1  5862  fvfullfunlem2  5863