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

Theorem conrel2d 41601
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
conrel2d (𝜑 → (𝐵𝐴) = ∅)

Proof of Theorem conrel2d
StepHypRef Expression
1 df-rn 5631 . . . . 5 ran 𝐴 = dom 𝐴
21ineq2i 4156 . . . 4 (dom 𝐵 ∩ ran 𝐴) = (dom 𝐵 ∩ dom 𝐴)
32a1i 11 . . 3 (𝜑 → (dom 𝐵 ∩ ran 𝐴) = (dom 𝐵 ∩ dom 𝐴))
4 conrel1d.a . . . . 5 (𝜑𝐴 = ∅)
54dmeqd 5847 . . . 4 (𝜑 → dom 𝐴 = dom ∅)
65ineq2d 4159 . . 3 (𝜑 → (dom 𝐵 ∩ dom 𝐴) = (dom 𝐵 ∩ dom ∅))
7 dm0 5862 . . . . . 6 dom ∅ = ∅
87ineq2i 4156 . . . . 5 (dom 𝐵 ∩ dom ∅) = (dom 𝐵 ∩ ∅)
9 in0 4338 . . . . 5 (dom 𝐵 ∩ ∅) = ∅
108, 9eqtri 2764 . . . 4 (dom 𝐵 ∩ dom ∅) = ∅
1110a1i 11 . . 3 (𝜑 → (dom 𝐵 ∩ dom ∅) = ∅)
123, 6, 113eqtrd 2780 . 2 (𝜑 → (dom 𝐵 ∩ ran 𝐴) = ∅)
1312coemptyd 14789 1 (𝜑 → (𝐵𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cin 3897  c0 4269  ccnv 5619  dom cdm 5620  ran crn 5621  ccom 5624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator