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Theorem cosnop 32704
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 4774 . . 3 (𝐴𝑉 → {𝐴} ≠ ∅)
3 xpco 6309 . . 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 5875 . 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 2785 1 (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}) = {⟨𝐶, 𝐵⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wne 2940  c0 4333  {csn 4626  cop 4632   × cxp 5683  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-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  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-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568
This theorem is referenced by:  coprprop  32708
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