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Theorem cosnopne 32703
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 6233 . . . . 5 (𝐵𝑊 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
31, 2syl 17 . . . 4 (𝜑 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
4 cosnopne.c . . . . 5 (𝜑𝐶𝑋)
5 rnsnopg 6241 . . . . 5 (𝐶𝑋 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
64, 5syl 17 . . . 4 (𝜑 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
73, 6ineq12d 4221 . . 3 (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ({𝐴} ∩ {𝐷}))
8 cosnopne.1 . . . 4 (𝜑𝐴𝐷)
9 disjsn2 4712 . . . 4 (𝐴𝐷 → ({𝐴} ∩ {𝐷}) = ∅)
108, 9syl 17 . . 3 (𝜑 → ({𝐴} ∩ {𝐷}) = ∅)
117, 10eqtrd 2777 . 2 (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ∅)
1211coemptyd 15018 1 (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wne 2940  cin 3950  c0 4333  {csn 4626  cop 4632  dom cdm 5685  ran crn 5686  ccom 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697
This theorem is referenced by:  coprprop  32708
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