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Theorem conrel1d 36974
 Description: Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
Hypothesis
Ref Expression
conrel1d.a (𝜑𝐴 = ∅)
Assertion
Ref Expression
conrel1d (𝜑 → (𝐴𝐵) = ∅)

Proof of Theorem conrel1d
StepHypRef Expression
1 incom 3767 . . 3 (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
2 dfdm4 5238 . . . . 5 dom 𝐴 = ran 𝐴
3 conrel1d.a . . . . . . 7 (𝜑𝐴 = ∅)
43rneqd 5274 . . . . . 6 (𝜑 → ran 𝐴 = ran ∅)
5 rn0 5298 . . . . . 6 ran ∅ = ∅
64, 5syl6eq 2660 . . . . 5 (𝜑 → ran 𝐴 = ∅)
72, 6syl5eq 2656 . . . 4 (𝜑 → dom 𝐴 = ∅)
8 ineq2 3770 . . . . 5 (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅))
9 in0 3920 . . . . 5 (ran 𝐵 ∩ ∅) = ∅
108, 9syl6eq 2660 . . . 4 (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅)
117, 10syl 17 . . 3 (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅)
121, 11syl5eq 2656 . 2 (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)
1312coemptyd 13566 1 (𝜑 → (𝐴𝐵) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∩ cin 3539  ∅c0 3874  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ∘ ccom 5042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050 This theorem is referenced by: (None)
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