Theorem cosnopne 33174

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	⊢ (𝜑 → 𝐵 ∈ 𝑊)
cosnopne.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)
cosnopne.1	⊢ (𝜑 → 𝐴 ≠ 𝐷)

Assertion

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

Proof of Theorem cosnopne

Step	Hyp	Ref	Expression
1		cosnopne.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2		dmsnopg 6213	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
4		cosnopne.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
5		rnsnopg 6221	. . . . 5 ⊢ (𝐶 ∈ 𝑋 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
6	4, 5	syl 18	. . . 4 ⊢ (𝜑 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
7	3, 6	ineq12d 4170	. . 3 ⊢ (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ({𝐴} ∩ {𝐷}))
8		cosnopne.1	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐷)
9		disjsn2 4676	. . . 4 ⊢ (𝐴 ≠ 𝐷 → ({𝐴} ∩ {𝐷}) = ∅)
10	8, 9	syl 18	. . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐷}) = ∅)
11	7, 10	eqtrd 2797	. 2 ⊢ (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ∅)
12	11	coemptyd 15056	1 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ≠ wne 2957   ∩ cin 3901  ∅c0 4282  {csn 4587  ⟨cop 4593  dom cdm 5659  ran crn 5660   ∘ ccom 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671

This theorem is used by:  coprprop  33179