Theorem cosnop 30431

Description: Composition of two ordered pair singletons with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)

Hypotheses

Ref	Expression
cosnop.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
cosnop.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
cosnop.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)

Assertion

Ref	Expression
cosnop	⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}) = {⟨𝐶, 𝐵⟩})

Proof of Theorem cosnop

Step	Hyp	Ref	Expression
1		cosnop.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		snnzg 4710	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
3		xpco 6140	. . 3 ⊢ ({𝐴} ≠ ∅ → (({𝐴} × {𝐵}) ∘ ({𝐶} × {𝐴})) = ({𝐶} × {𝐵}))
4	1, 2, 3	3syl 18	. 2 ⊢ (𝜑 → (({𝐴} × {𝐵}) ∘ ({𝐶} × {𝐴})) = ({𝐶} × {𝐵}))
5		cosnop.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6		xpsng 6901	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
7	1, 5, 6	syl2anc 586	. . 3 ⊢ (𝜑 → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
8		cosnop.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
9		xpsng 6901	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → ({𝐶} × {𝐴}) = {⟨𝐶, 𝐴⟩})
10	8, 1, 9	syl2anc 586	. . 3 ⊢ (𝜑 → ({𝐶} × {𝐴}) = {⟨𝐶, 𝐴⟩})
11	7, 10	coeq12d 5735	. 2 ⊢ (𝜑 → (({𝐴} × {𝐵}) ∘ ({𝐶} × {𝐴})) = ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}))
12		xpsng 6901	. . 3 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → ({𝐶} × {𝐵}) = {⟨𝐶, 𝐵⟩})
13	8, 5, 12	syl2anc 586	. 2 ⊢ (𝜑 → ({𝐶} × {𝐵}) = {⟨𝐶, 𝐵⟩})
14	4, 11, 13	3eqtr3d 2864	1 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}) = {⟨𝐶, 𝐵⟩})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∅c0 4291  {csn 4567  ⟨cop 4573   × cxp 5553   ∘ ccom 5559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362

This theorem is referenced by:  coprprop  30435