Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  conrel2d Structured version   Visualization version   Unicode version

Theorem conrel2d 36256
Description: Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
Hypothesis
Ref Expression
conrel1d.a  |-  ( ph  ->  `' A  =  (/) )
Assertion
Ref Expression
conrel2d  |-  ( ph  ->  ( B  o.  A
)  =  (/) )

Proof of Theorem conrel2d
StepHypRef Expression
1 df-rn 4845 . . . . 5  |-  ran  A  =  dom  `' A
21ineq2i 3631 . . . 4  |-  ( dom 
B  i^i  ran  A )  =  ( dom  B  i^i  dom  `' A )
32a1i 11 . . 3  |-  ( ph  ->  ( dom  B  i^i  ran 
A )  =  ( dom  B  i^i  dom  `' A ) )
4 conrel1d.a . . . . 5  |-  ( ph  ->  `' A  =  (/) )
54dmeqd 5037 . . . 4  |-  ( ph  ->  dom  `' A  =  dom  (/) )
65ineq2d 3634 . . 3  |-  ( ph  ->  ( dom  B  i^i  dom  `' A )  =  ( dom  B  i^i  dom  (/) ) )
7 dm0 5048 . . . . . 6  |-  dom  (/)  =  (/)
87ineq2i 3631 . . . . 5  |-  ( dom 
B  i^i  dom  (/) )  =  ( dom  B  i^i  (/) )
9 in0 3760 . . . . 5  |-  ( dom 
B  i^i  (/) )  =  (/)
108, 9eqtri 2473 . . . 4  |-  ( dom 
B  i^i  dom  (/) )  =  (/)
1110a1i 11 . . 3  |-  ( ph  ->  ( dom  B  i^i  dom  (/) )  =  (/) )
123, 6, 113eqtrd 2489 . 2  |-  ( ph  ->  ( dom  B  i^i  ran 
A )  =  (/) )
1312coemptyd 13043 1  |-  ( ph  ->  ( B  o.  A
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1444    i^i cin 3403   (/)c0 3731   `'ccnv 4833   dom cdm 4834   ran crn 4835    o. ccom 4838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator