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Theorem cosnop 32710
Description: Composition of two ordered pair singletons with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
Hypotheses
Ref Expression
cosnop.a (𝜑𝐴𝑉)
cosnop.b (𝜑𝐵𝑊)
cosnop.c (𝜑𝐶𝑋)
Assertion
Ref Expression
cosnop (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}) = {⟨𝐶, 𝐵⟩})

Proof of Theorem cosnop
StepHypRef Expression
1 cosnop.a . . 3 (𝜑𝐴𝑉)
2 snnzg 4779 . . 3 (𝐴𝑉 → {𝐴} ≠ ∅)
3 xpco 6311 . . 3 ({𝐴} ≠ ∅ → (({𝐴} × {𝐵}) ∘ ({𝐶} × {𝐴})) = ({𝐶} × {𝐵}))
41, 2, 33syl 18 . 2 (𝜑 → (({𝐴} × {𝐵}) ∘ ({𝐶} × {𝐴})) = ({𝐶} × {𝐵}))
5 cosnop.b . . . 4 (𝜑𝐵𝑊)
6 xpsng 7159 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
71, 5, 6syl2anc 584 . . 3 (𝜑 → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
8 cosnop.c . . . 4 (𝜑𝐶𝑋)
9 xpsng 7159 . . . 4 ((𝐶𝑋𝐴𝑉) → ({𝐶} × {𝐴}) = {⟨𝐶, 𝐴⟩})
108, 1, 9syl2anc 584 . . 3 (𝜑 → ({𝐶} × {𝐴}) = {⟨𝐶, 𝐴⟩})
117, 10coeq12d 5878 . 2 (𝜑 → (({𝐴} × {𝐵}) ∘ ({𝐶} × {𝐴})) = ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}))
12 xpsng 7159 . . 3 ((𝐶𝑋𝐵𝑊) → ({𝐶} × {𝐵}) = {⟨𝐶, 𝐵⟩})
138, 5, 12syl2anc 584 . 2 (𝜑 → ({𝐶} × {𝐵}) = {⟨𝐶, 𝐵⟩})
144, 11, 133eqtr3d 2783 1 (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}) = {⟨𝐶, 𝐵⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wne 2938  c0 4339  {csn 4631  cop 4637   × cxp 5687  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-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  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-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by:  coprprop  32714
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