ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  resmpt3 Unicode version

Theorem resmpt3 4600
Description: Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)
Assertion
Ref Expression
resmpt3  |->  C  |`  i^i 
|->  C
Distinct variable groups:   ,   ,
Allowed substitution hint:    C()

Proof of Theorem resmpt3
StepHypRef Expression
1 resres 4567 . 2  |->  C  |`  |`  |->  C  |`  i^i
2 ssid 2958 . . . 4  C_
3 resmpt 4599 . . . 4 
C_  |->  C  |`  |->  C
42, 3ax-mp 7 . . 3  |->  C  |`  |->  C
54reseq1i 4551 . 2  |->  C  |`  |`  |->  C  |`
6 inss1 3151 . . 3  i^i  C_
7 resmpt 4599 . . 3  i^i 
C_  |->  C  |`  i^i  i^i  |->  C
86, 7ax-mp 7 . 2  |->  C  |`  i^i  i^i 
|->  C
91, 5, 83eqtr3i 2065 1  |->  C  |`  i^i 
|->  C
Colors of variables: wff set class
Syntax hints:   wceq 1242    i^i cin 2910    C_ wss 2911    |-> cmpt 3809    |` cres 4290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-mpt 3811  df-xp 4294  df-rel 4295  df-res 4300
This theorem is referenced by:  offres  5704
  Copyright terms: Public domain W3C validator