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Theorem conrel1d 40840
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 4092 . . 3 (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
2 dfdm4 5739 . . . . 5 dom 𝐴 = ran 𝐴
3 conrel1d.a . . . . . . 7 (𝜑𝐴 = ∅)
43rneqd 5782 . . . . . 6 (𝜑 → ran 𝐴 = ran ∅)
5 rn0 5770 . . . . . 6 ran ∅ = ∅
64, 5eqtrdi 2790 . . . . 5 (𝜑 → ran 𝐴 = ∅)
72, 6syl5eq 2786 . . . 4 (𝜑 → dom 𝐴 = ∅)
8 ineq2 4098 . . . . 5 (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅))
9 in0 4281 . . . . 5 (ran 𝐵 ∩ ∅) = ∅
108, 9eqtrdi 2790 . . . 4 (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅)
117, 10syl 17 . . 3 (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅)
121, 11syl5eq 2786 . 2 (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)
1312coemptyd 14431 1 (𝜑 → (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cin 3843  c0 4212  ccnv 5525  dom cdm 5526  ran crn 5527  ccom 5530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-br 5032  df-opab 5094  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538
This theorem is referenced by: (None)
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