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Theorem elfix 5787
Description: Membership in the fixed points of a relationship. (Contributed by SF, 11-Feb-2015.)
Assertion
Ref Expression
elfix (A Fix RARA)

Proof of Theorem elfix
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2867 . 2 (A Fix RA V)
2 brex 4689 . . 3 (ARA → (A V A V))
32simpld 445 . 2 (ARAA V)
4 eleq1 2413 . . 3 (x = A → (x Fix RA Fix R))
5 breq12 4644 . . . 4 ((x = A x = A) → (xRxARA))
65anidms 626 . . 3 (x = A → (xRxARA))
7 df-fix 5740 . . . . 5 Fix R = ran (R ∩ I )
87eleq2i 2417 . . . 4 (x Fix Rx ran (R ∩ I ))
9 elrn 4896 . . . . 5 (x ran (R ∩ I ) ↔ y y(R ∩ I )x)
10 brin 4693 . . . . . . 7 (y(R ∩ I )x ↔ (yRx y I x))
11 ancom 437 . . . . . . 7 ((yRx y I x) ↔ (y I x yRx))
12 vex 2862 . . . . . . . . 9 x V
1312ideq 4870 . . . . . . . 8 (y I xy = x)
1413anbi1i 676 . . . . . . 7 ((y I x yRx) ↔ (y = x yRx))
1510, 11, 143bitri 262 . . . . . 6 (y(R ∩ I )x ↔ (y = x yRx))
1615exbii 1582 . . . . 5 (y y(R ∩ I )xy(y = x yRx))
179, 16bitri 240 . . . 4 (x ran (R ∩ I ) ↔ y(y = x yRx))
18 breq1 4642 . . . . 5 (y = x → (yRxxRx))
1912, 18ceqsexv 2894 . . . 4 (y(y = x yRx) ↔ xRx)
208, 17, 193bitri 262 . . 3 (x Fix RxRx)
214, 6, 20vtoclbg 2915 . 2 (A V → (A Fix RARA))
221, 3, 21pm5.21nii 342 1 (A Fix RARA)
Colors of variables: wff setvar class
Syntax hints:  wb 176   wa 358  wex 1541   = wceq 1642   wcel 1710  Vcvv 2859  cin 3208   class class class wbr 4639   I cid 4763  ran crn 4773   Fix cfix 5739
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-ima 4727  df-id 4767  df-rn 4786  df-fix 5740
This theorem is referenced by:  epprc  5827  clos1ex  5876  nnltp1clem1  6261  addccan2nclem2  6264  nchoicelem10  6298
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