Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  conrel1d Structured version   Visualization version   GIF version

Theorem conrel1d 43676
Description: Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
Hypothesis
Ref Expression
conrel1d.a (𝜑𝐴 = ∅)
Assertion
Ref Expression
conrel1d (𝜑 → (𝐴𝐵) = ∅)

Proof of Theorem conrel1d
StepHypRef Expression
1 incom 4209 . . 3 (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
2 dfdm4 5906 . . . . 5 dom 𝐴 = ran 𝐴
3 conrel1d.a . . . . . . 7 (𝜑𝐴 = ∅)
43rneqd 5949 . . . . . 6 (𝜑 → ran 𝐴 = ran ∅)
5 rn0 5936 . . . . . 6 ran ∅ = ∅
64, 5eqtrdi 2793 . . . . 5 (𝜑 → ran 𝐴 = ∅)
72, 6eqtrid 2789 . . . 4 (𝜑 → dom 𝐴 = ∅)
8 ineq2 4214 . . . . 5 (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅))
9 in0 4395 . . . . 5 (ran 𝐵 ∩ ∅) = ∅
108, 9eqtrdi 2793 . . . 4 (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅)
117, 10syl 17 . . 3 (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅)
121, 11eqtrid 2789 . 2 (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)
1312coemptyd 15018 1 (𝜑 → (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cin 3950  c0 4333  ccnv 5684  dom cdm 5685  ran crn 5686  ccom 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator