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Theorem cosnopne 32709
Description: Composition of two ordered pair singletons with non-matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
Hypotheses
Ref Expression
cosnopne.b (𝜑𝐵𝑊)
cosnopne.c (𝜑𝐶𝑋)
cosnopne.1 (𝜑𝐴𝐷)
Assertion
Ref Expression
cosnopne (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)

Proof of Theorem cosnopne
StepHypRef Expression
1 cosnopne.b . . . . 5 (𝜑𝐵𝑊)
2 dmsnopg 6235 . . . . 5 (𝐵𝑊 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
31, 2syl 17 . . . 4 (𝜑 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
4 cosnopne.c . . . . 5 (𝜑𝐶𝑋)
5 rnsnopg 6243 . . . . 5 (𝐶𝑋 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
64, 5syl 17 . . . 4 (𝜑 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
73, 6ineq12d 4229 . . 3 (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ({𝐴} ∩ {𝐷}))
8 cosnopne.1 . . . 4 (𝜑𝐴𝐷)
9 disjsn2 4717 . . . 4 (𝐴𝐷 → ({𝐴} ∩ {𝐷}) = ∅)
108, 9syl 17 . . 3 (𝜑 → ({𝐴} ∩ {𝐷}) = ∅)
117, 10eqtrd 2775 . 2 (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ∅)
1211coemptyd 15015 1 (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wne 2938  cin 3962  c0 4339  {csn 4631  cop 4637  dom cdm 5689  ran crn 5690  ccom 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701
This theorem is referenced by:  coprprop  32714
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