Theorem cosnopne 30428

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 6063	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
4		cosnopne.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
5		rnsnopg 6071	. . . . 5 ⊢ (𝐶 ∈ 𝑋 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
6	4, 5	syl 17	. . . 4 ⊢ (𝜑 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
7	3, 6	ineq12d 4183	. . 3 ⊢ (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ({𝐴} ∩ {𝐷}))
8		cosnopne.1	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐷)
9		disjsn2 4641	. . . 4 ⊢ (𝐴 ≠ 𝐷 → ({𝐴} ∩ {𝐷}) = ∅)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐷}) = ∅)
11	7, 10	eqtrd 2855	. 2 ⊢ (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ∅)
12	11	coemptyd 14332	1 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113   ≠ wne 3015   ∩ cin 3928  ∅c0 4284  {csn 4560  ⟨cop 4566  dom cdm 5548  ran crn 5549   ∘ ccom 5552

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-br 5060  df-opab 5122  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560

This theorem is referenced by:  coprprop  30433