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Theorem disjrnmpt 27656
Description: Rewriting a disjoint collection using the range of a mapping. (Contributed by Thierry Arnoux, 27-May-2020.)
Assertion
Ref Expression
disjrnmpt  |-  (Disj  x  e.  A  B  -> Disj  y  e.  ran  ( x  e.  A  |->  B ) y )
Distinct variable groups:    x, A, y    y, B
Allowed substitution hint:    B( x)

Proof of Theorem disjrnmpt
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 disjabrex 27653 . 2  |-  (Disj  x  e.  A  B  -> Disj  y  e.  { z  |  E. x  e.  A  z  =  B } y )
2 eqid 2454 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
32rnmpt 5237 . . 3  |-  ran  (
x  e.  A  |->  B )  =  { z  |  E. x  e.  A  z  =  B }
4 disjeq1 4417 . . 3  |-  ( ran  ( x  e.  A  |->  B )  =  {
z  |  E. x  e.  A  z  =  B }  ->  (Disj  y  e.  ran  ( x  e.  A  |->  B ) y  <-> Disj  y  e.  { z  |  E. x  e.  A  z  =  B }
y ) )
53, 4ax-mp 5 . 2  |-  (Disj  y  e.  ran  ( x  e.  A  |->  B ) y  <-> Disj  y  e.  { z  |  E. x  e.  A  z  =  B }
y )
61, 5sylibr 212 1  |-  (Disj  x  e.  A  B  -> Disj  y  e.  ran  ( x  e.  A  |->  B ) y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398   {cab 2439   E.wrex 2805  Disj wdisj 4410    |-> cmpt 4497   ran crn 4989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-cnv 4996  df-dm 4998  df-rn 4999
This theorem is referenced by:  sigapildsys  28388  carsgclctunlem2  28527
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