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Theorem cosnopne 30498
 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 6041 . . . . 5 (𝐵𝑊 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
31, 2syl 17 . . . 4 (𝜑 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
4 cosnopne.c . . . . 5 (𝜑𝐶𝑋)
5 rnsnopg 6049 . . . . 5 (𝐶𝑋 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
64, 5syl 17 . . . 4 (𝜑 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
73, 6ineq12d 4143 . . 3 (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ({𝐴} ∩ {𝐷}))
8 cosnopne.1 . . . 4 (𝜑𝐴𝐷)
9 disjsn2 4611 . . . 4 (𝐴𝐷 → ({𝐴} ∩ {𝐷}) = ∅)
108, 9syl 17 . . 3 (𝜑 → ({𝐴} ∩ {𝐷}) = ∅)
117, 10eqtrd 2833 . 2 (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ∅)
1211coemptyd 14350 1 (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ∩ cin 3882  ∅c0 4246  {csn 4528  ⟨cop 4534  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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  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:  coprprop  30503
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