MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvmpt2v Unicode version

Theorem cbvmpt2v 6052
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4212, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
cbvmpt2v.1  |-  ( x  =  z  ->  C  =  E )
cbvmpt2v.2  |-  ( y  =  w  ->  E  =  D )
Assertion
Ref Expression
cbvmpt2v  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  A ,  w  e.  B  |->  D )
Distinct variable groups:    x, w, y, z, A    w, B, x, y, z    w, C, z    x, D, y
Allowed substitution hints:    C( x, y)    D( z, w)    E( x, y, z, w)

Proof of Theorem cbvmpt2v
StepHypRef Expression
1 nfcv 2502 . 2  |-  F/_ z C
2 nfcv 2502 . 2  |-  F/_ w C
3 nfcv 2502 . 2  |-  F/_ x D
4 nfcv 2502 . 2  |-  F/_ y D
5 cbvmpt2v.1 . . 3  |-  ( x  =  z  ->  C  =  E )
6 cbvmpt2v.2 . . 3  |-  ( y  =  w  ->  E  =  D )
75, 6sylan9eq 2418 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
81, 2, 3, 4, 7cbvmpt2 6051 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  A ,  w  e.  B  |->  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. cmpt2 5983
This theorem is referenced by:  seqomlem0  6603  dffi3  7331  cantnfsuc  7518  fin23lem33  8118  om2uzrdg  11183  uzrdgsuci  11187  sadcp1  12854  smupp1  12879  imasvscafn  13649  sylow1  15124  sylow2b  15144  sylow3lem5  15152  sylow3  15154  efgmval  15231  efgtf  15241  txbas  17479  bcth  18966  opnmbl  19172  mbfimaopn  19226  mbfi1fseq  19291  opsqrlem3  23156  dya2iocival  24207  sxbrsigalem5  24222  sxbrsigalem6  24223  cvmliftlem15  24553  cvmlift2  24571  sdc  26046  tendoplcbv  31023  dvhvaddcbv  31338  dvhvscacbv  31347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-opab 4180  df-oprab 5985  df-mpt2 5986
  Copyright terms: Public domain W3C validator