NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  otsnelsi3 GIF version

Theorem otsnelsi3 5805
Description: Ordered triple membership in triple singleton image. (Contributed by SF, 12-Feb-2015.)
Hypotheses
Ref Expression
otsnelsi3.1 A V
otsnelsi3.2 B V
otsnelsi3.3 C V
Assertion
Ref Expression
otsnelsi3 ({A}, {B}, {C} SI3 RA, B, C R)

Proof of Theorem otsnelsi3
Dummy variables p x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-si3 5758 . . 3 SI3 R = (( SI 1st ⊗ ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))) “ 1R)
21eleq2i 2417 . 2 ({A}, {B}, {C} SI3 R{A}, {B}, {C} (( SI 1st ⊗ ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))) “ 1R))
3 elimapw1 4944 . 2 ({A}, {B}, {C} (( SI 1st ⊗ ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))) “ 1R) ↔ p R {p}, {A}, {B}, {C} ( SI 1st ⊗ ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))))
4 oteltxp 5782 . . . . 5 ({p}, {A}, {B}, {C} ( SI 1st ⊗ ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))) ↔ ({p}, {A} SI 1st {p}, {B}, {C} ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))))
5 vex 2862 . . . . . . . 8 p V
6 otsnelsi3.1 . . . . . . . 8 A V
75, 6opsnelsi 5774 . . . . . . 7 ({p}, {A} SI 1stp, A 1st )
8 df-br 4640 . . . . . . 7 (p1st Ap, A 1st )
97, 8bitr4i 243 . . . . . 6 ({p}, {A} SI 1stp1st A)
10 oteltxp 5782 . . . . . . 7 ({p}, {B}, {C} ( SI (1st 2nd ) ⊗ SI (2nd 2nd )) ↔ ({p}, {B} SI (1st 2nd ) {p}, {C} SI (2nd 2nd )))
11 otsnelsi3.2 . . . . . . . . . 10 B V
125, 11opsnelsi 5774 . . . . . . . . 9 ({p}, {B} SI (1st 2nd ) ↔ p, B (1st 2nd ))
13 opelco 4884 . . . . . . . . 9 (p, B (1st 2nd ) ↔ x(p2nd x x1st B))
14 opeq 4619 . . . . . . . . . . . . . 14 p = Proj1 p, Proj2 p
1514breq1i 4646 . . . . . . . . . . . . 13 (p2nd x Proj1 p, Proj2 p2nd x)
165proj1ex 4593 . . . . . . . . . . . . . 14 Proj1 p V
175proj2ex 4594 . . . . . . . . . . . . . 14 Proj2 p V
1816, 17opbr2nd 5502 . . . . . . . . . . . . 13 ( Proj1 p, Proj2 p2nd x Proj2 p = x)
19 eqcom 2355 . . . . . . . . . . . . 13 ( Proj2 p = xx = Proj2 p)
2015, 18, 193bitri 262 . . . . . . . . . . . 12 (p2nd xx = Proj2 p)
2120anbi1i 676 . . . . . . . . . . 11 ((p2nd x x1st B) ↔ (x = Proj2 p x1st B))
2221exbii 1582 . . . . . . . . . 10 (x(p2nd x x1st B) ↔ x(x = Proj2 p x1st B))
23 breq1 4642 . . . . . . . . . . 11 (x = Proj2 p → (x1st B Proj2 p1st B))
2417, 23ceqsexv 2894 . . . . . . . . . 10 (x(x = Proj2 p x1st B) ↔ Proj2 p1st B)
2522, 24bitri 240 . . . . . . . . 9 (x(p2nd x x1st B) ↔ Proj2 p1st B)
2612, 13, 253bitri 262 . . . . . . . 8 ({p}, {B} SI (1st 2nd ) ↔ Proj2 p1st B)
27 otsnelsi3.3 . . . . . . . . . 10 C V
285, 27opsnelsi 5774 . . . . . . . . 9 ({p}, {C} SI (2nd 2nd ) ↔ p, C (2nd 2nd ))
29 opelco 4884 . . . . . . . . 9 (p, C (2nd 2nd ) ↔ x(p2nd x x2nd C))
3020anbi1i 676 . . . . . . . . . . 11 ((p2nd x x2nd C) ↔ (x = Proj2 p x2nd C))
3130exbii 1582 . . . . . . . . . 10 (x(p2nd x x2nd C) ↔ x(x = Proj2 p x2nd C))
32 breq1 4642 . . . . . . . . . . 11 (x = Proj2 p → (x2nd C Proj2 p2nd C))
3317, 32ceqsexv 2894 . . . . . . . . . 10 (x(x = Proj2 p x2nd C) ↔ Proj2 p2nd C)
3431, 33bitri 240 . . . . . . . . 9 (x(p2nd x x2nd C) ↔ Proj2 p2nd C)
3528, 29, 343bitri 262 . . . . . . . 8 ({p}, {C} SI (2nd 2nd ) ↔ Proj2 p2nd C)
3626, 35anbi12i 678 . . . . . . 7 (({p}, {B} SI (1st 2nd ) {p}, {C} SI (2nd 2nd )) ↔ ( Proj2 p1st B Proj2 p2nd C))
3716, 17opbr2nd 5502 . . . . . . . 8 ( Proj1 p, Proj2 p2nd B, C Proj2 p = B, C)
3814breq1i 4646 . . . . . . . 8 (p2nd B, C Proj1 p, Proj2 p2nd B, C)
3911, 27op1st2nd 5790 . . . . . . . 8 (( Proj2 p1st B Proj2 p2nd C) ↔ Proj2 p = B, C)
4037, 38, 393bitr4ri 269 . . . . . . 7 (( Proj2 p1st B Proj2 p2nd C) ↔ p2nd B, C)
4110, 36, 403bitri 262 . . . . . 6 ({p}, {B}, {C} ( SI (1st 2nd ) ⊗ SI (2nd 2nd )) ↔ p2nd B, C)
429, 41anbi12i 678 . . . . 5 (({p}, {A} SI 1st {p}, {B}, {C} ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))) ↔ (p1st A p2nd B, C))
4311, 27opex 4588 . . . . . 6 B, C V
446, 43op1st2nd 5790 . . . . 5 ((p1st A p2nd B, C) ↔ p = A, B, C)
454, 42, 443bitri 262 . . . 4 ({p}, {A}, {B}, {C} ( SI 1st ⊗ ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))) ↔ p = A, B, C)
4645rexbii 2639 . . 3 (p R {p}, {A}, {B}, {C} ( SI 1st ⊗ ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))) ↔ p R p = A, B, C)
47 risset 2661 . . 3 (A, B, C Rp R p = A, B, C)
4846, 47bitr4i 243 . 2 (p R {p}, {A}, {B}, {C} ( SI 1st ⊗ ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))) ↔ A, B, C R)
492, 3, 483bitri 262 1 ({A}, {B}, {C} SI3 RA, B, C R)
Colors of variables: wff setvar class
Syntax hints:  wb 176   wa 358  wex 1541   = wceq 1642   wcel 1710  wrex 2615  Vcvv 2859  {csn 3737  1cpw1 4135  cop 4561   Proj1 cproj1 4563   Proj2 cproj2 4564   class class class wbr 4639  1st c1st 4717   SI csi 4720   ccom 4721  cima 4722  2nd c2nd 4783  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
  Copyright terms: Public domain W3C validator