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Theorem conrel2d 40362
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
conrel2d (𝜑 → (𝐵𝐴) = ∅)

Proof of Theorem conrel2d
StepHypRef Expression
1 df-rn 5534 . . . . 5 ran 𝐴 = dom 𝐴
21ineq2i 4139 . . . 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 5742 . . . 4 (𝜑 → dom 𝐴 = dom ∅)
65ineq2d 4142 . . 3 (𝜑 → (dom 𝐵 ∩ dom 𝐴) = (dom 𝐵 ∩ dom ∅))
7 dm0 5758 . . . . . 6 dom ∅ = ∅
87ineq2i 4139 . . . . 5 (dom 𝐵 ∩ dom ∅) = (dom 𝐵 ∩ ∅)
9 in0 4302 . . . . 5 (dom 𝐵 ∩ ∅) = ∅
108, 9eqtri 2824 . . . 4 (dom 𝐵 ∩ dom ∅) = ∅
1110a1i 11 . . 3 (𝜑 → (dom 𝐵 ∩ dom ∅) = ∅)
123, 6, 113eqtrd 2840 . 2 (𝜑 → (dom 𝐵 ∩ ran 𝐴) = ∅)
1312coemptyd 14334 1 (𝜑 → (𝐵𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  cin 3883  c0 4246  ccnv 5522  dom cdm 5523  ran crn 5524  ccom 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535
This theorem is referenced by: (None)
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