Theorem otsnelsi3 5805

Description: Ordered triple membership in triple singleton image. (Contributed by SF, 12-Feb-2015.)

Hypotheses

Ref	Expression
otsnelsi3.1	|- A e. _V
otsnelsi3.2	|- B e. _V
otsnelsi3.3	|- C e. _V

Assertion

Ref	Expression
otsnelsi3	|- (<.{A}, <.{B}, {C}>.>. e. SI3 R <-> <.A, <.B, C>.>. e. R)

Proof of Theorem otsnelsi3

Dummy variables p x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-si3 5758	. . 3 |- SI3 R = (( SI 1st (x) ( SI (1st o. 2nd) (x) SI (2nd o. 2nd)))"~P1 R)
2	1	eleq2i 2417	. 2 |- (<.{A}, <.{B}, {C}>.>. e. SI3 R <-> <.{A}, <.{B}, {C}>.>. e. (( SI 1st (x) ( SI (1st o. 2nd) (x) SI (2nd o. 2nd)))"~P1 R))
3		elimapw1 4944	. 2 |- (<.{A}, <.{B}, {C}>.>. e. (( SI 1st (x) ( SI (1st o. 2nd) (x) SI (2nd o. 2nd)))"~P1 R) <-> E.p e. R <.{p}, <.{A}, <.{B}, {C}>.>.>. e. ( SI 1st (x) ( SI (1st o. 2nd) (x) SI (2nd o. 2nd))))
4		oteltxp 5782	. . . . 5 |- (<.{p}, <.{A}, <.{B}, {C}>.>.>. e. ( SI 1st (x) ( SI (1st o. 2nd) (x) SI (2nd o. 2nd))) <-> (<.{p}, {A}>. e. SI 1st /\ <.{p}, <.{B}, {C}>.>. e. ( SI (1st o. 2nd) (x) SI (2nd o. 2nd))))
5		vex 2862	. . . . . . . 8 |- p e. _V
6		otsnelsi3.1	. . . . . . . 8 |- A e. _V
7	5, 6	opsnelsi 5774	. . . . . . 7 |- (<.{p}, {A}>. e. SI 1st <-> <.p, A>. e. 1st)
8		df-br 4640	. . . . . . 7 |- (p1stA <-> <.p, A>. e. 1st)
9	7, 8	bitr4i 243	. . . . . 6 |- (<.{p}, {A}>. e. SI 1st <-> p1stA)
10		oteltxp 5782	. . . . . . 7 |- (<.{p}, <.{B}, {C}>.>. e. ( SI (1st o. 2nd) (x) SI (2nd o. 2nd)) <-> (<.{p}, {B}>. e. SI (1st o. 2nd) /\ <.{p}, {C}>. e. SI (2nd o. 2nd)))
11		otsnelsi3.2	. . . . . . . . . 10 |- B e. _V
12	5, 11	opsnelsi 5774	. . . . . . . . 9 |- (<.{p}, {B}>. e. SI (1st o. 2nd) <-> <.p, B>. e. (1st o. 2nd))
13		opelco 4884	. . . . . . . . 9 |- (<.p, B>. e. (1st o. 2nd) <-> E.x(p2ndx /\ x1stB))
14		opeq 4619	. . . . . . . . . . . . . 14 |- p = <. Proj1 p, Proj2 p>.
15	14	breq1i 4646	. . . . . . . . . . . . 13 |- (p2ndx <-> <. Proj1 p, Proj2 p>.2ndx)
16	5	proj1ex 4593	. . . . . . . . . . . . . 14 |- Proj1 p e. _V
17	5	proj2ex 4594	. . . . . . . . . . . . . 14 |- Proj2 p e. _V
18	16, 17	opbr2nd 5502	. . . . . . . . . . . . 13 |- (<. Proj1 p, Proj2 p>.2ndx <-> Proj2 p = x)
19		eqcom 2355	. . . . . . . . . . . . 13 |- ( Proj2 p = x <-> x = Proj2 p)
20	15, 18, 19	3bitri 262	. . . . . . . . . . . 12 |- (p2ndx <-> x = Proj2 p)
21	20	anbi1i 676	. . . . . . . . . . 11 |- ((p2ndx /\ x1stB) <-> (x = Proj2 p /\ x1stB))
22	21	exbii 1582	. . . . . . . . . 10 |- (E.x(p2ndx /\ x1stB) <-> E.x(x = Proj2 p /\ x1stB))
23		breq1 4642	. . . . . . . . . . 11 |- (x = Proj2 p -> (x1stB <-> Proj2 p1stB))
24	17, 23	ceqsexv 2894	. . . . . . . . . 10 |- (E.x(x = Proj2 p /\ x1stB) <-> Proj2 p1stB)
25	22, 24	bitri 240	. . . . . . . . 9 |- (E.x(p2ndx /\ x1stB) <-> Proj2 p1stB)
26	12, 13, 25	3bitri 262	. . . . . . . 8 |- (<.{p}, {B}>. e. SI (1st o. 2nd) <-> Proj2 p1stB)
27		otsnelsi3.3	. . . . . . . . . 10 |- C e. _V
28	5, 27	opsnelsi 5774	. . . . . . . . 9 |- (<.{p}, {C}>. e. SI (2nd o. 2nd) <-> <.p, C>. e. (2nd o. 2nd))
29		opelco 4884	. . . . . . . . 9 |- (<.p, C>. e. (2nd o. 2nd) <-> E.x(p2ndx /\ x2ndC))
30	20	anbi1i 676	. . . . . . . . . . 11 |- ((p2ndx /\ x2ndC) <-> (x = Proj2 p /\ x2ndC))
31	30	exbii 1582	. . . . . . . . . 10 |- (E.x(p2ndx /\ x2ndC) <-> E.x(x = Proj2 p /\ x2ndC))
32		breq1 4642	. . . . . . . . . . 11 |- (x = Proj2 p -> (x2ndC <-> Proj2 p2ndC))
33	17, 32	ceqsexv 2894	. . . . . . . . . 10 |- (E.x(x = Proj2 p /\ x2ndC) <-> Proj2 p2ndC)
34	31, 33	bitri 240	. . . . . . . . 9 |- (E.x(p2ndx /\ x2ndC) <-> Proj2 p2ndC)
35	28, 29, 34	3bitri 262	. . . . . . . 8 |- (<.{p}, {C}>. e. SI (2nd o. 2nd) <-> Proj2 p2ndC)
36	26, 35	anbi12i 678	. . . . . . 7 |- ((<.{p}, {B}>. e. SI (1st o. 2nd) /\ <.{p}, {C}>. e. SI (2nd o. 2nd)) <-> ( Proj2 p1stB /\ Proj2 p2ndC))
37	16, 17	opbr2nd 5502	. . . . . . . 8 |- (<. Proj1 p, Proj2 p>.2nd<.B, C>. <-> Proj2 p = <.B, C>.)
38	14	breq1i 4646	. . . . . . . 8 |- (p2nd<.B, C>. <-> <. Proj1 p, Proj2 p>.2nd<.B, C>.)
39	11, 27	op1st2nd 5790	. . . . . . . 8 |- (( Proj2 p1stB /\ Proj2 p2ndC) <-> Proj2 p = <.B, C>.)
40	37, 38, 39	3bitr4ri 269	. . . . . . 7 |- (( Proj2 p1stB /\ Proj2 p2ndC) <-> p2nd<.B, C>.)
41	10, 36, 40	3bitri 262	. . . . . 6 |- (<.{p}, <.{B}, {C}>.>. e. ( SI (1st o. 2nd) (x) SI (2nd o. 2nd)) <-> p2nd<.B, C>.)
42	9, 41	anbi12i 678	. . . . 5 |- ((<.{p}, {A}>. e. SI 1st /\ <.{p}, <.{B}, {C}>.>. e. ( SI (1st o. 2nd) (x) SI (2nd o. 2nd))) <-> (p1stA /\ p2nd<.B, C>.))
43	11, 27	opex 4588	. . . . . 6 |- <.B, C>. e. _V
44	6, 43	op1st2nd 5790	. . . . 5 |- ((p1stA /\ p2nd<.B, C>.) <-> p = <.A, <.B, C>.>.)
45	4, 42, 44	3bitri 262	. . . 4 |- (<.{p}, <.{A}, <.{B}, {C}>.>.>. e. ( SI 1st (x) ( SI (1st o. 2nd) (x) SI (2nd o. 2nd))) <-> p = <.A, <.B, C>.>.)
46	45	rexbii 2639	. . 3 |- (E.p e. R <.{p}, <.{A}, <.{B}, {C}>.>.>. e. ( SI 1st (x) ( SI (1st o. 2nd) (x) SI (2nd o. 2nd))) <-> E.p e. R p = <.A, <.B, C>.>.)
47		risset 2661	. . 3 |- (<.A, <.B, C>.>. e. R <-> E.p e. R p = <.A, <.B, C>.>.)
48	46, 47	bitr4i 243	. 2 |- (E.p e. R <.{p}, <.{A}, <.{B}, {C}>.>.>. e. ( SI 1st (x) ( SI (1st o. 2nd) (x) SI (2nd o. 2nd))) <-> <.A, <.B, C>.>. e. R)
49	2, 3, 48	3bitri 262	1 |- (<.{A}, <.{B}, {C}>.>. e. SI3 R <-> <.A, <.B, C>.>. e. R)

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2615  _Vcvv 2859  {csn 3737  ~P₁ cpw1 4135  <.cop 4561   Proj1 cproj1 4563   Proj2 cproj2 4564   class class class wbr 4639  1stc1st 4717   SI csi 4720   o. ccom 4721  "cima 4722  2ndc2nd 4783   (x) ctxp 5735   SI₃ csi3 5757

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 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-1st 4723  df-co 4726  df-ima 4727  df-si 4728  df-cnv 4785  df-2nd 4797  df-txp 5736  df-si3 5758

This theorem is referenced by:  composeex  5820  addcfnex  5824  funsex  5828  crossex  5850  domfnex  5870  ranfnex  5871  transex  5910  antisymex  5912  connexex  5913  foundex  5914  extex  5915  symex  5916  mucex  6133  ovcelem1  6171  ceex  6174