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Theorem conrel2d 37423
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
conrel2d (𝜑 → (𝐵𝐴) = ∅)

Proof of Theorem conrel2d
StepHypRef Expression
1 df-rn 5090 . . . . 5 ran 𝐴 = dom 𝐴
21ineq2i 3794 . . . 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 5291 . . . 4 (𝜑 → dom 𝐴 = dom ∅)
65ineq2d 3797 . . 3 (𝜑 → (dom 𝐵 ∩ dom 𝐴) = (dom 𝐵 ∩ dom ∅))
7 dm0 5303 . . . . . 6 dom ∅ = ∅
87ineq2i 3794 . . . . 5 (dom 𝐵 ∩ dom ∅) = (dom 𝐵 ∩ ∅)
9 in0 3945 . . . . 5 (dom 𝐵 ∩ ∅) = ∅
108, 9eqtri 2648 . . . 4 (dom 𝐵 ∩ dom ∅) = ∅
1110a1i 11 . . 3 (𝜑 → (dom 𝐵 ∩ dom ∅) = ∅)
123, 6, 113eqtrd 2664 . 2 (𝜑 → (dom 𝐵 ∩ ran 𝐴) = ∅)
1312coemptyd 13647 1 (𝜑 → (𝐵𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  cin 3559  c0 3896  ccnv 5078  dom cdm 5079  ran crn 5080  ccom 5083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091
This theorem is referenced by: (None)
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