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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-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-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
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