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Theorem conrel1d 43124
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 4203 . . 3 (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
2 dfdm4 5902 . . . . 5 dom 𝐴 = ran 𝐴
3 conrel1d.a . . . . . . 7 (𝜑𝐴 = ∅)
43rneqd 5944 . . . . . 6 (𝜑 → ran 𝐴 = ran ∅)
5 rn0 5932 . . . . . 6 ran ∅ = ∅
64, 5eqtrdi 2784 . . . . 5 (𝜑 → ran 𝐴 = ∅)
72, 6eqtrid 2780 . . . 4 (𝜑 → dom 𝐴 = ∅)
8 ineq2 4208 . . . . 5 (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅))
9 in0 4395 . . . . 5 (ran 𝐵 ∩ ∅) = ∅
108, 9eqtrdi 2784 . . . 4 (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅)
117, 10syl 17 . . 3 (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅)
121, 11eqtrid 2780 . 2 (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)
1312coemptyd 14966 1 (𝜑 → (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cin 3948  c0 4326  ccnv 5681  dom cdm 5682  ran crn 5683  ccom 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694
This theorem is referenced by: (None)
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