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Theorem conrel2d 41916
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 5643 . . . . 5 ran 𝐴 = dom 𝐴
21ineq2i 4168 . . . 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 5860 . . . 4 (𝜑 → dom 𝐴 = dom ∅)
65ineq2d 4171 . . 3 (𝜑 → (dom 𝐵 ∩ dom 𝐴) = (dom 𝐵 ∩ dom ∅))
7 dm0 5875 . . . . . 6 dom ∅ = ∅
87ineq2i 4168 . . . . 5 (dom 𝐵 ∩ dom ∅) = (dom 𝐵 ∩ ∅)
9 in0 4350 . . . . 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 14861 1 (𝜑 → (𝐵𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  cin 3908  c0 4281  ccnv 5631  dom cdm 5632  ran crn 5633  ccom 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644
This theorem is referenced by: (None)
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