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Theorem conrel1d 43093
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 4201 . . 3 (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
2 dfdm4 5898 . . . . 5 dom 𝐴 = ran 𝐴
3 conrel1d.a . . . . . . 7 (𝜑𝐴 = ∅)
43rneqd 5940 . . . . . 6 (𝜑 → ran 𝐴 = ran ∅)
5 rn0 5928 . . . . . 6 ran ∅ = ∅
64, 5eqtrdi 2784 . . . . 5 (𝜑 → ran 𝐴 = ∅)
72, 6eqtrid 2780 . . . 4 (𝜑 → dom 𝐴 = ∅)
8 ineq2 4206 . . . . 5 (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅))
9 in0 4392 . . . . 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 14959 1 (𝜑 → (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  cin 3946  c0 4323  ccnv 5677  dom cdm 5678  ran crn 5679  ccom 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690
This theorem is referenced by: (None)
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